svm.cpp

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/*============================================================================

libsvm

Copyright (c) 2000-2019 Chih-Chung Chang and Chih-Jen Lin
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:

1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.

2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.

3. Neither name of copyright holders nor the names of its contributors
may be used to endorse or promote products derived from this software
without specific prior written permission.


THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR
CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

============================================================================*/

#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cctype>
#include <cfloat>
#include <cstring>
#include <cstdarg>
#include <climits>
#include <clocale>
#include <mitkLocaleSwitch.h>
#include "svm.h"
int libsvm_version = LIBSVM_VERSION;
typedef float Qfloat;
typedef signed char schar;
#ifndef min
template <class T> static inline T min(T x,T y) { return (x<y)?x:y; }
#endif
#ifndef max
template <class T> static inline T max(T x,T y) { return (x>y)?x:y; }
#endif
template <class T> static inline void swap(T& x, T& y) { T t=x; x=y; y=t; }
template <class S, class T> static inline void clone(T*& dst, S* src, int n)
{
  dst = new T[n];
  memcpy((void *)dst,(void *)src,sizeof(T)*n);
}
static inline double powi(double base, int times)
{
  double tmp = base, ret = 1.0;

  for(int t=times; t>0; t/=2)
  {
    if(t%2==1) ret*=tmp;
    tmp = tmp * tmp;
  }
  return ret;
}
#define INF HUGE_VAL
#define TAU 1e-12
#define Malloc(type,n) (type *)malloc((n)*sizeof(type))

static void print_string_stdout(const char *s)
{
  fputs(s,stdout);
  fflush(stdout);
}
static void (*svm_print_string) (const char *) = &print_string_stdout;
#if 1
static void info(const char *fmt,...)
{
  char buf[BUFSIZ];
  va_list ap;
  va_start(ap,fmt);
  vsprintf(buf,fmt,ap);
  va_end(ap);
  (*svm_print_string)(buf);
}
#else
static void info(const char *fmt,...) {}
#endif

//
// Kernel Cache
//
// l is the number of total data items
// size is the cache size limit in bytes
//
class Cache
{
public:
  Cache(int l,long int size);
  ~Cache();

  // request data [0,len)
  // return some position p where [p,len) need to be filled
  // (p >= len if nothing needs to be filled)
  int get_data(const int index, Qfloat **data, int len);
  void swap_index(int i, int j);
private:
  int l;
  long int size;
  struct head_t
  {
    head_t *prev, *next; // a circular list
    Qfloat *data;
    int len;  // data[0,len) is cached in this entry
  };

  head_t *head;
  head_t lru_head;
  void lru_delete(head_t *h);
  void lru_insert(head_t *h);
};

Cache::Cache(int l_,long int size_):l(l_),size(size_)
{
  head = (head_t *)calloc(l,sizeof(head_t)); // initialized to 0
  size /= sizeof(Qfloat);
  size -= l * sizeof(head_t) / sizeof(Qfloat);
  size = max(size, 2 * (long int) l); // cache must be large enough for two columns
  lru_head.next = lru_head.prev = &lru_head;
}

Cache::~Cache()
{
  for(head_t *h = lru_head.next; h != &lru_head; h=h->next)
    free(h->data);
  free(head);
}

void Cache::lru_delete(head_t *h)
{
  // delete from current location
  h->prev->next = h->next;
  h->next->prev = h->prev;
}

void Cache::lru_insert(head_t *h)
{
  // insert to last position
  h->next = &lru_head;
  h->prev = lru_head.prev;
  h->prev->next = h;
  h->next->prev = h;
}

int Cache::get_data(const int index, Qfloat **data, int len)
{
  head_t *h = &head[index];
  if(h->len) lru_delete(h);
  int more = len - h->len;

  if(more > 0)
  {
    // free old space
    while(size < more)
    {
      head_t *old = lru_head.next;
      lru_delete(old);
      free(old->data);
      size += old->len;
      old->data = nullptr;
      old->len = 0;
    }

    // allocate new space
    h->data = (Qfloat *)realloc(h->data,sizeof(Qfloat)*len);
    size -= more;
    swap(h->len,len);
  }

  lru_insert(h);
  *data = h->data;
  return len;
}

void Cache::swap_index(int i, int j)
{
  if(i==j) return;

  if(head[i].len) lru_delete(&head[i]);
  if(head[j].len) lru_delete(&head[j]);
  swap(head[i].data,head[j].data);
  swap(head[i].len,head[j].len);
  if(head[i].len) lru_insert(&head[i]);
  if(head[j].len) lru_insert(&head[j]);

  if(i>j) swap(i,j);
  for(head_t *h = lru_head.next; h!=&lru_head; h=h->next)
  {
    if(h->len > i)
    {
      if(h->len > j)
        swap(h->data[i],h->data[j]);
      else
      {
        // give up
        lru_delete(h);
        free(h->data);
        size += h->len;
        h->data = nullptr;
        h->len = 0;
      }
    }
  }
}

//
// Kernel evaluation
//
// the static method k_function is for doing single kernel evaluation
// the constructor of Kernel prepares to calculate the l*l kernel matrix
// the member function get_Q is for getting one column from the Q Matrix
//
class QMatrix {
public:
  virtual Qfloat *get_Q(int column, int len) const = 0;
  virtual double *get_QD() const = 0;
  virtual void swap_index(int i, int j) const = 0;
  virtual ~QMatrix() {}
};

class Kernel: public QMatrix {
public:
  Kernel(int l, svm_node * const * x, const svm_parameter& param);
  ~Kernel() override;

  static double k_function(const svm_node *x, const svm_node *y,
                           const svm_parameter& param);
  Qfloat *get_Q(int column, int len) const override = 0;
  double *get_QD() const override = 0;
  void swap_index(int i, int j) const override // no so const...
  {
    swap(x[i],x[j]);
    if(x_square) swap(x_square[i],x_square[j]);
  }
protected:

  double (Kernel::*kernel_function)(int i, int j) const;

private:
  const svm_node **x;
  double *x_square;

  // svm_parameter
  const int kernel_type;
  const int degree;
  const double gamma;
  const double coef0;

  static double dot(const svm_node *px, const svm_node *py);
  double kernel_linear(int i, int j) const
  {
    return dot(x[i],x[j]);
  }
  double kernel_poly(int i, int j) const
  {
    return powi(gamma*dot(x[i],x[j])+coef0,degree);
  }
  double kernel_rbf(int i, int j) const
  {
    return exp(-gamma*(x_square[i]+x_square[j]-2*dot(x[i],x[j])));
  }
  double kernel_sigmoid(int i, int j) const
  {
    return tanh(gamma*dot(x[i],x[j])+coef0);
  }
  double kernel_precomputed(int i, int j) const
  {
    return x[i][(int)(x[j][0].value)].value;
  }
};

Kernel::Kernel(int l, svm_node * const * x_, const svm_parameter& param)
  :kernel_type(param.kernel_type), degree(param.degree),
    gamma(param.gamma), coef0(param.coef0)
{
  switch(kernel_type)
  {
  case LINEAR:
    kernel_function = &Kernel::kernel_linear;
    break;
  case POLY:
    kernel_function = &Kernel::kernel_poly;
    break;
  case RBF:
    kernel_function = &Kernel::kernel_rbf;
    break;
  case SIGMOID:
    kernel_function = &Kernel::kernel_sigmoid;
    break;
  case PRECOMPUTED:
    kernel_function = &Kernel::kernel_precomputed;
    break;
  }

  clone(x,x_,l);

  if(kernel_type == RBF)
  {
    x_square = new double[l];
    for(int i=0;i<l;i++)
      x_square[i] = dot(x[i],x[i]);
  }
  else
    x_square = nullptr;
}

Kernel::~Kernel()
{
  delete[] x;
  delete[] x_square;
}

double Kernel::dot(const svm_node *px, const svm_node *py)
{
  double sum = 0;
  while(px->index != -1 && py->index != -1)
  {
    if(px->index == py->index)
    {
      sum += px->value * py->value;
      ++px;
      ++py;
    }
    else
    {
      if(px->index > py->index)
        ++py;
      else
        ++px;
    }
  }
  return sum;
}

double Kernel::k_function(const svm_node *x, const svm_node *y,
                          const svm_parameter& param)
{
  switch(param.kernel_type)
  {
  case LINEAR:
    return dot(x,y);
  case POLY:
    return powi(param.gamma*dot(x,y)+param.coef0,param.degree);
  case RBF:
  {
    double sum = 0;
    while(x->index != -1 && y->index !=-1)
    {
      if(x->index == y->index)
      {
        double d = x->value - y->value;
        sum += d*d;
        ++x;
        ++y;
      }
      else
      {
        if(x->index > y->index)
        {
          sum += y->value * y->value;
          ++y;
        }
        else
        {
          sum += x->value * x->value;
          ++x;
        }
      }
    }

    while(x->index != -1)
    {
      sum += x->value * x->value;
      ++x;
    }

    while(y->index != -1)
    {
      sum += y->value * y->value;
      ++y;
    }

    return exp(-param.gamma*sum);
  }
  case SIGMOID:
    return tanh(param.gamma*dot(x,y)+param.coef0);
  case PRECOMPUTED:  //x: test (validation), y: SV
    return x[(int)(y->value)].value;
  default:
    return 0;  // Unreachable
  }
}

// An SMO algorithm in Fan et al., JMLR 6(2005), p. 1889--1918
// Solves:
//
// min 0.5(\alpha^T Q \alpha) + p^T \alpha
//
//  y^T \alpha = \delta
//  y_i = +1 or -1
//  0 <= alpha_i <= Cp for y_i = 1
//  0 <= alpha_i <= Cn for y_i = -1
//
// Given:
//
// Q, p, y, Cp, Cn, and an initial feasible point \alpha
// l is the size of vectors and matrices
// eps is the stopping tolerance
//
// solution will be put in \alpha, objective value will be put in obj
//
class Solver {
public:
  Solver() {};
  virtual ~Solver() {};

  struct SolutionInfo {
    double obj;
    double rho;
    double *upper_bound;
    double r; // for Solver_NU
  };

  void Solve(int l, const QMatrix& Q, const double *p_, const schar *y_,
             double *alpha_, const double* C_, double eps,
             SolutionInfo* si, int shrinking);
protected:
  int active_size;
  schar *y;
  double *G;  // gradient of objective function
  enum { LOWER_BOUND, UPPER_BOUND, FREE };
  char *alpha_status; // LOWER_BOUND, UPPER_BOUND, FREE
  double *alpha;
  const QMatrix *Q;
  const double *QD;
  double eps;
  double Cp,Cn;
  double *C;
  double *p;
  int *active_set;
  double *G_bar;  // gradient, if we treat free variables as 0
  int l;
  bool unshrink; // XXX

  double get_C(int i)
  {
    return C[i];
  }
  void update_alpha_status(int i)
  {
    if(alpha[i] >= get_C(i))
      alpha_status[i] = UPPER_BOUND;
    else if(alpha[i] <= 0)
      alpha_status[i] = LOWER_BOUND;
    else alpha_status[i] = FREE;
  }
  bool is_upper_bound(int i) { return alpha_status[i] == UPPER_BOUND; }
  bool is_lower_bound(int i) { return alpha_status[i] == LOWER_BOUND; }
  bool is_free(int i) { return alpha_status[i] == FREE; }
  void swap_index(int i, int j);
  void reconstruct_gradient();
  virtual int select_working_set(int &i, int &j);
  virtual double calculate_rho();
  virtual void do_shrinking();
private:
  bool be_shrunk(int i, double Gmax1, double Gmax2);
};

void Solver::swap_index(int i, int j)
{
  Q->swap_index(i,j);
  swap(y[i],y[j]);
  swap(G[i],G[j]);
  swap(alpha_status[i],alpha_status[j]);
  swap(alpha[i],alpha[j]);
  swap(p[i],p[j]);
  swap(active_set[i],active_set[j]);
  swap(G_bar[i],G_bar[j]);
  swap(C[i],C[j]);
}

void Solver::reconstruct_gradient()
{
  // reconstruct inactive elements of G from G_bar and free variables

  if(active_size == l) return;

  int i,j;
  int nr_free = 0;

  for(j=active_size;j<l;j++)
    G[j] = G_bar[j] + p[j];

  for(j=0;j<active_size;j++)
    if(is_free(j))
      nr_free++;

  if(2*nr_free < active_size)
    info("\nWARNING: using -h 0 may be faster\n");

  if (nr_free*l > 2*active_size*(l-active_size))
  {
    for(i=active_size;i<l;i++)
    {
      const Qfloat *Q_i = Q->get_Q(i,active_size);
      for(j=0;j<active_size;j++)
        if(is_free(j))
          G[i] += alpha[j] * Q_i[j];
    }
  }
  else
  {
    for(i=0;i<active_size;i++)
      if(is_free(i))
      {
        const Qfloat *Q_i = Q->get_Q(i,l);
        double alpha_i = alpha[i];
        for(j=active_size;j<l;j++)
          G[j] += alpha_i * Q_i[j];
      }
  }
}

void Solver::Solve(int l, const QMatrix& Q, const double *p_, const schar *y_,
                   double *alpha_, const double* C_, double eps,
                   SolutionInfo* si, int shrinking)
{
  this->l = l;
  this->Q = &Q;
  QD=Q.get_QD();
  clone(p, p_,l);
  clone(y, y_,l);
  clone(alpha,alpha_,l);
  clone(C,C_,l);
  this->eps = eps;
  unshrink = false;

  // initialize alpha_status
  {
    alpha_status = new char[l];
    for(int i=0;i<l;i++)
      update_alpha_status(i);
  }

  // initialize active set (for shrinking)
  {
    active_set = new int[l];
    for(int i=0;i<l;i++)
      active_set[i] = i;
    active_size = l;
  }

  // initialize gradient
  {
    G = new double[l];
    G_bar = new double[l];
    int i;
    for(i=0;i<l;i++)
    {
      G[i] = p[i];
      G_bar[i] = 0;
    }
    for(i=0;i<l;i++)
      if(!is_lower_bound(i))
      {
        const Qfloat *Q_i = Q.get_Q(i,l);
        double alpha_i = alpha[i];
        int j;
        for(j=0;j<l;j++)
          G[j] += alpha_i*Q_i[j];
        if(is_upper_bound(i))
          for(j=0;j<l;j++)
            G_bar[j] += get_C(i) * Q_i[j];
      }
  }

  // optimization step

  int iter = 0;
  int max_iter = max(10000000, l>INT_MAX/100 ? INT_MAX : 100*l);
  int counter = min(l,1000)+1;

  while(iter < max_iter)
  {
    // show progress and do shrinking

    if(--counter == 0)
    {
      counter = min(l,1000);
      if(shrinking) do_shrinking();
      info(".");
    }

    int i,j;
    if(select_working_set(i,j)!=0)
    {
      // reconstruct the whole gradient
      reconstruct_gradient();
      // reset active set size and check
      active_size = l;
      info("*");
      if(select_working_set(i,j)!=0)
        break;
      else
        counter = 1; // do shrinking next iteration
    }

    ++iter;

    // update alpha[i] and alpha[j], handle bounds carefully

    const Qfloat *Q_i = Q.get_Q(i,active_size);
    const Qfloat *Q_j = Q.get_Q(j,active_size);

    double C_i = get_C(i);
    double C_j = get_C(j);

    double old_alpha_i = alpha[i];
    double old_alpha_j = alpha[j];

    if(y[i]!=y[j])
    {
      double quad_coef = QD[i]+QD[j]+2*Q_i[j];
      if (quad_coef <= 0)
        quad_coef = TAU;
      double delta = (-G[i]-G[j])/quad_coef;
      double diff = alpha[i] - alpha[j];
      alpha[i] += delta;
      alpha[j] += delta;

      if(diff > 0)
      {
        if(alpha[j] < 0)
        {
          alpha[j] = 0;
          alpha[i] = diff;
        }
      }
      else
      {
        if(alpha[i] < 0)
        {
          alpha[i] = 0;
          alpha[j] = -diff;
        }
      }
      if(diff > C_i - C_j)
      {
        if(alpha[i] > C_i)
        {
          alpha[i] = C_i;
          alpha[j] = C_i - diff;
        }
      }
      else
      {
        if(alpha[j] > C_j)
        {
          alpha[j] = C_j;
          alpha[i] = C_j + diff;
        }
      }
    }
    else
    {
      double quad_coef = QD[i]+QD[j]-2*Q_i[j];
      if (quad_coef <= 0)
        quad_coef = TAU;
      double delta = (G[i]-G[j])/quad_coef;
      double sum = alpha[i] + alpha[j];
      alpha[i] -= delta;
      alpha[j] += delta;

      if(sum > C_i)
      {
        if(alpha[i] > C_i)
        {
          alpha[i] = C_i;
          alpha[j] = sum - C_i;
        }
      }
      else
      {
        if(alpha[j] < 0)
        {
          alpha[j] = 0;
          alpha[i] = sum;
        }
      }
      if(sum > C_j)
      {
        if(alpha[j] > C_j)
        {
          alpha[j] = C_j;
          alpha[i] = sum - C_j;
        }
      }
      else
      {
        if(alpha[i] < 0)
        {
          alpha[i] = 0;
          alpha[j] = sum;
        }
      }
    }

    // update G

    double delta_alpha_i = alpha[i] - old_alpha_i;
    double delta_alpha_j = alpha[j] - old_alpha_j;

    for(int k=0;k<active_size;k++)
    {
      G[k] += Q_i[k]*delta_alpha_i + Q_j[k]*delta_alpha_j;
    }

    // update alpha_status and G_bar

    {
      bool ui = is_upper_bound(i);
      bool uj = is_upper_bound(j);
      update_alpha_status(i);
      update_alpha_status(j);
      int k;
      if(ui != is_upper_bound(i))
      {
        Q_i = Q.get_Q(i,l);
        if(ui)
          for(k=0;k<l;k++)
            G_bar[k] -= C_i * Q_i[k];
        else
          for(k=0;k<l;k++)
            G_bar[k] += C_i * Q_i[k];
      }

      if(uj != is_upper_bound(j))
      {
        Q_j = Q.get_Q(j,l);
        if(uj)
          for(k=0;k<l;k++)
            G_bar[k] -= C_j * Q_j[k];
        else
          for(k=0;k<l;k++)
            G_bar[k] += C_j * Q_j[k];
      }
    }
  }

  if(iter >= max_iter)
  {
    if(active_size < l)
    {
      // reconstruct the whole gradient to calculate objective value
      reconstruct_gradient();
      active_size = l;
      info("*");
    }
    fprintf(stderr,"\nWARNING: reaching max number of iterations\n");
  }

  // calculate rho

  si->rho = calculate_rho();

  // calculate objective value
  {
    double v = 0;
    int i;
    for(i=0;i<l;i++)
      v += alpha[i] * (G[i] + p[i]);

    si->obj = v/2;
  }

  // put back the solution
  {
    for(int i=0;i<l;i++)
      alpha_[active_set[i]] = alpha[i];
  }

  // juggle everything back
  /*{
  for(int i=0;i<l;i++)
   while(active_set[i] != i)
    swap_index(i,active_set[i]);
    // or Q.swap_index(i,active_set[i]);
 }*/

  for(int i=0;i<l;i++)
    si->upper_bound[i] = C[i];

  info("\noptimization finished, #iter = %d\n",iter);

  delete[] p;
  delete[] y;
  delete[] C;
  delete[] alpha;
  delete[] alpha_status;
  delete[] active_set;
  delete[] G;
  delete[] G_bar;
}

// return 1 if already optimal, return 0 otherwise
int Solver::select_working_set(int &out_i, int &out_j)
{
  // return i,j such that
  // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha)
  // j: minimizes the decrease of obj value
  //    (if quadratic coefficeint <= 0, replace it with tau)
  //    -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha)

  double Gmax = -INF;
  double Gmax2 = -INF;
  int Gmax_idx = -1;
  int Gmin_idx = -1;
  double obj_diff_min = INF;

  for(int t=0;t<active_size;t++)
    if(y[t]==+1)
    {
      if(!is_upper_bound(t))
        if(-G[t] >= Gmax)
        {
          Gmax = -G[t];
          Gmax_idx = t;
        }
    }
    else
    {
      if(!is_lower_bound(t))
        if(G[t] >= Gmax)
        {
          Gmax = G[t];
          Gmax_idx = t;
        }
    }

  int i = Gmax_idx;
  const Qfloat *Q_i = nullptr;
  if(i != -1) // nullptr Q_i not accessed: Gmax=-INF if i=-1
    Q_i = Q->get_Q(i,active_size);

  for(int j=0;j<active_size;j++)
  {
    if(y[j]==+1)
    {
      if (!is_lower_bound(j))
      {
        double grad_diff=Gmax+G[j];
        if (G[j] >= Gmax2)
          Gmax2 = G[j];
        if (grad_diff > 0)
        {
          double obj_diff;
          double quad_coef = QD[i]+QD[j]-2.0*y[i]*Q_i[j];
          if (quad_coef > 0)
            obj_diff = -(grad_diff*grad_diff)/quad_coef;
          else
            obj_diff = -(grad_diff*grad_diff)/TAU;

          if (obj_diff <= obj_diff_min)
          {
            Gmin_idx=j;
            obj_diff_min = obj_diff;
          }
        }
      }
    }
    else
    {
      if (!is_upper_bound(j))
      {
        double grad_diff= Gmax-G[j];
        if (-G[j] >= Gmax2)
          Gmax2 = -G[j];
        if (grad_diff > 0)
        {
          double obj_diff;
          double quad_coef = QD[i]+QD[j]+2.0*y[i]*Q_i[j];
          if (quad_coef > 0)
            obj_diff = -(grad_diff*grad_diff)/quad_coef;
          else
            obj_diff = -(grad_diff*grad_diff)/TAU;

          if (obj_diff <= obj_diff_min)
          {
            Gmin_idx=j;
            obj_diff_min = obj_diff;
          }
        }
      }
    }
  }

  if(Gmax+Gmax2 < eps)
    return 1;

  out_i = Gmax_idx;
  out_j = Gmin_idx;
  return 0;
}

bool Solver::be_shrunk(int i, double Gmax1, double Gmax2)
{
  if(is_upper_bound(i))
  {
    if(y[i]==+1)
      return(-G[i] > Gmax1);
    else
      return(-G[i] > Gmax2);
  }
  else if(is_lower_bound(i))
  {
    if(y[i]==+1)
      return(G[i] > Gmax2);
    else
      return(G[i] > Gmax1);
  }
  else
    return(false);
}

void Solver::do_shrinking()
{
  int i;
  double Gmax1 = -INF;  // max { -y_i * grad(f)_i | i in I_up(\alpha) }
  double Gmax2 = -INF;  // max { y_i * grad(f)_i | i in I_low(\alpha) }

  // find maximal violating pair first
  for(i=0;i<active_size;i++)
  {
    if(y[i]==+1)
    {
      if(!is_upper_bound(i))
      {
        if(-G[i] >= Gmax1)
          Gmax1 = -G[i];
      }
      if(!is_lower_bound(i))
      {
        if(G[i] >= Gmax2)
          Gmax2 = G[i];
      }
    }
    else
    {
      if(!is_upper_bound(i))
      {
        if(-G[i] >= Gmax2)
          Gmax2 = -G[i];
      }
      if(!is_lower_bound(i))
      {
        if(G[i] >= Gmax1)
          Gmax1 = G[i];
      }
    }
  }

  if(unshrink == false && Gmax1 + Gmax2 <= eps*10)
  {
    unshrink = true;
    reconstruct_gradient();
    active_size = l;
    info("*");
  }

  for(i=0;i<active_size;i++)
    if (be_shrunk(i, Gmax1, Gmax2))
    {
      active_size--;
      while (active_size > i)
      {
        if (!be_shrunk(active_size, Gmax1, Gmax2))
        {
          swap_index(i,active_size);
          break;
        }
        active_size--;
      }
    }
}

double Solver::calculate_rho()
{
  double r;
  int nr_free = 0;
  double ub = INF, lb = -INF, sum_free = 0;
  for(int i=0;i<active_size;i++)
  {
    double yG = y[i]*G[i];

    if(is_upper_bound(i))
    {
      if(y[i]==-1)
        ub = min(ub,yG);
      else
        lb = max(lb,yG);
    }
    else if(is_lower_bound(i))
    {
      if(y[i]==+1)
        ub = min(ub,yG);
      else
        lb = max(lb,yG);
    }
    else
    {
      ++nr_free;
      sum_free += yG;
    }
  }

  if(nr_free>0)
    r = sum_free/nr_free;
  else
    r = (ub+lb)/2;

  return r;
}

//
// Solver for nu-svm classification and regression
//
// additional constraint: e^T \alpha = constant
//
class Solver_NU: public Solver
{
public:
  Solver_NU() {}
  void Solve(int l, const QMatrix& Q, const double *p, const schar *y,
             double *alpha, double* C_, double eps,
             SolutionInfo* si, int shrinking)
  {
    this->si = si;
    Solver::Solve(l,Q,p,y,alpha,C_,eps,si,shrinking);
  }
private:
  SolutionInfo *si;
  int select_working_set(int &i, int &j) override;
  double calculate_rho() override;
  bool be_shrunk(int i, double Gmax1, double Gmax2, double Gmax3, double Gmax4);
  void do_shrinking() override;
};

// return 1 if already optimal, return 0 otherwise
int Solver_NU::select_working_set(int &out_i, int &out_j)
{
  // return i,j such that y_i = y_j and
  // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha)
  // j: minimizes the decrease of obj value
  //    (if quadratic coefficeint <= 0, replace it with tau)
  //    -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha)

  double Gmaxp = -INF;
  double Gmaxp2 = -INF;
  int Gmaxp_idx = -1;

  double Gmaxn = -INF;
  double Gmaxn2 = -INF;
  int Gmaxn_idx = -1;

  int Gmin_idx = -1;
  double obj_diff_min = INF;

  for(int t=0;t<active_size;t++)
    if(y[t]==+1)
    {
      if(!is_upper_bound(t))
        if(-G[t] >= Gmaxp)
        {
          Gmaxp = -G[t];
          Gmaxp_idx = t;
        }
    }
    else
    {
      if(!is_lower_bound(t))
        if(G[t] >= Gmaxn)
        {
          Gmaxn = G[t];
          Gmaxn_idx = t;
        }
    }

  int ip = Gmaxp_idx;
  int in = Gmaxn_idx;
  const Qfloat *Q_ip = nullptr;
  const Qfloat *Q_in = nullptr;
  if(ip != -1) // nullptr Q_ip not accessed: Gmaxp=-INF if ip=-1
    Q_ip = Q->get_Q(ip,active_size);
  if(in != -1)
    Q_in = Q->get_Q(in,active_size);

  for(int j=0;j<active_size;j++)
  {
    if(y[j]==+1)
    {
      if (!is_lower_bound(j))
      {
        double grad_diff=Gmaxp+G[j];
        if (G[j] >= Gmaxp2)
          Gmaxp2 = G[j];
        if (grad_diff > 0)
        {
          double obj_diff;
          double quad_coef = QD[ip]+QD[j]-2*Q_ip[j];
          if (quad_coef > 0)
            obj_diff = -(grad_diff*grad_diff)/quad_coef;
          else
            obj_diff = -(grad_diff*grad_diff)/TAU;

          if (obj_diff <= obj_diff_min)
          {
            Gmin_idx=j;
            obj_diff_min = obj_diff;
          }
        }
      }
    }
    else
    {
      if (!is_upper_bound(j))
      {
        double grad_diff=Gmaxn-G[j];
        if (-G[j] >= Gmaxn2)
          Gmaxn2 = -G[j];
        if (grad_diff > 0)
        {
          double obj_diff;
          double quad_coef = QD[in]+QD[j]-2*Q_in[j];
          if (quad_coef > 0)
            obj_diff = -(grad_diff*grad_diff)/quad_coef;
          else
            obj_diff = -(grad_diff*grad_diff)/TAU;

          if (obj_diff <= obj_diff_min)
          {
            Gmin_idx=j;
            obj_diff_min = obj_diff;
          }
        }
      }
    }
  }

  if(max(Gmaxp+Gmaxp2,Gmaxn+Gmaxn2) < eps)
    return 1;

  if (y[Gmin_idx] == +1)
    out_i = Gmaxp_idx;
  else
    out_i = Gmaxn_idx;
  out_j = Gmin_idx;

  return 0;
}

bool Solver_NU::be_shrunk(int i, double Gmax1, double Gmax2, double Gmax3, double Gmax4)
{
  if(is_upper_bound(i))
  {
    if(y[i]==+1)
      return(-G[i] > Gmax1);
    else
      return(-G[i] > Gmax4);
  }
  else if(is_lower_bound(i))
  {
    if(y[i]==+1)
      return(G[i] > Gmax2);
    else
      return(G[i] > Gmax3);
  }
  else
    return(false);
}

void Solver_NU::do_shrinking()
{
  double Gmax1 = -INF; // max { -y_i * grad(f)_i | y_i = +1, i in I_up(\alpha) }
  double Gmax2 = -INF; // max { y_i * grad(f)_i | y_i = +1, i in I_low(\alpha) }
  double Gmax3 = -INF; // max { -y_i * grad(f)_i | y_i = -1, i in I_up(\alpha) }
  double Gmax4 = -INF; // max { y_i * grad(f)_i | y_i = -1, i in I_low(\alpha) }

  // find maximal violating pair first
  int i;
  for(i=0;i<active_size;i++)
  {
    if(!is_upper_bound(i))
    {
      if(y[i]==+1)
      {
        if(-G[i] > Gmax1) Gmax1 = -G[i];
      }
      else if(-G[i] > Gmax4) Gmax4 = -G[i];
    }
    if(!is_lower_bound(i))
    {
      if(y[i]==+1)
      {
        if(G[i] > Gmax2) Gmax2 = G[i];
      }
      else if(G[i] > Gmax3) Gmax3 = G[i];
    }
  }

  if(unshrink == false && max(Gmax1+Gmax2,Gmax3+Gmax4) <= eps*10)
  {
    unshrink = true;
    reconstruct_gradient();
    active_size = l;
  }

  for(i=0;i<active_size;i++)
    if (be_shrunk(i, Gmax1, Gmax2, Gmax3, Gmax4))
    {
      active_size--;
      while (active_size > i)
      {
        if (!be_shrunk(active_size, Gmax1, Gmax2, Gmax3, Gmax4))
        {
          swap_index(i,active_size);
          break;
        }
        active_size--;
      }
    }
}

double Solver_NU::calculate_rho()
{
  int nr_free1 = 0,nr_free2 = 0;
  double ub1 = INF, ub2 = INF;
  double lb1 = -INF, lb2 = -INF;
  double sum_free1 = 0, sum_free2 = 0;

  for(int i=0;i<active_size;i++)
  {
    if(y[i]==+1)
    {
      if(is_upper_bound(i))
        lb1 = max(lb1,G[i]);
      else if(is_lower_bound(i))
        ub1 = min(ub1,G[i]);
      else
      {
        ++nr_free1;
        sum_free1 += G[i];
      }
    }
    else
    {
      if(is_upper_bound(i))
        lb2 = max(lb2,G[i]);
      else if(is_lower_bound(i))
        ub2 = min(ub2,G[i]);
      else
      {
        ++nr_free2;
        sum_free2 += G[i];
      }
    }
  }

  double r1,r2;
  if(nr_free1 > 0)
    r1 = sum_free1/nr_free1;
  else
    r1 = (ub1+lb1)/2;

  if(nr_free2 > 0)
    r2 = sum_free2/nr_free2;
  else
    r2 = (ub2+lb2)/2;

  si->r = (r1+r2)/2;
  return (r1-r2)/2;
}

//
// Q matrices for various formulations
//
class SVC_Q: public Kernel
{
public:
  SVC_Q(const svm_problem& prob, const svm_parameter& param, const schar *y_)
    :Kernel(prob.l, prob.x, param)
  {
    clone(y,y_,prob.l);
    cache = new Cache(prob.l,(long int)(param.cache_size*(1<<20)));
    QD = new double[prob.l];
    for(int i=0;i<prob.l;i++)
      QD[i] = (this->*kernel_function)(i,i);
  }

  Qfloat *get_Q(int i, int len) const override
  {
    Qfloat *data;
    int start, j;
    if((start = cache->get_data(i,&data,len)) < len)
    {
      for(j=start;j<len;j++)
        data[j] = (Qfloat)(y[i]*y[j]*(this->*kernel_function)(i,j));
    }
    return data;
  }

  double *get_QD() const override
  {
    return QD;
  }

  void swap_index(int i, int j) const override
  {
    cache->swap_index(i,j);
    Kernel::swap_index(i,j);
    swap(y[i],y[j]);
    swap(QD[i],QD[j]);
  }

  ~SVC_Q() override
  {
    delete[] y;
    delete cache;
    delete[] QD;
  }
private:
  schar *y;
  Cache *cache;
  double *QD;
};

class ONE_CLASS_Q: public Kernel
{
public:
  ONE_CLASS_Q(const svm_problem& prob, const svm_parameter& param)
    :Kernel(prob.l, prob.x, param)
  {
    cache = new Cache(prob.l,(long int)(param.cache_size*(1<<20)));
    QD = new double[prob.l];
    for(int i=0;i<prob.l;i++)
      QD[i] = (this->*kernel_function)(i,i);
  }

  Qfloat *get_Q(int i, int len) const override
  {
    Qfloat *data;
    int start, j;
    if((start = cache->get_data(i,&data,len)) < len)
    {
      for(j=start;j<len;j++)
        data[j] = (Qfloat)(this->*kernel_function)(i,j);
    }
    return data;
  }

  double *get_QD() const override
  {
    return QD;
  }

  void swap_index(int i, int j) const override
  {
    cache->swap_index(i,j);
    Kernel::swap_index(i,j);
    swap(QD[i],QD[j]);
  }

  ~ONE_CLASS_Q() override
  {
    delete cache;
    delete[] QD;
  }
private:
  Cache *cache;
  double *QD;
};

class SVR_Q: public Kernel
{
public:
  SVR_Q(const svm_problem& prob, const svm_parameter& param)
    :Kernel(prob.l, prob.x, param)
  {
    l = prob.l;
    cache = new Cache(l,(long int)(param.cache_size*(1<<20)));
    QD = new double[2*l];
    sign = new schar[2*l];
    index = new int[2*l];
    for(int k=0;k<l;k++)
    {
      sign[k] = 1;
      sign[k+l] = -1;
      index[k] = k;
      index[k+l] = k;
      QD[k] = (this->*kernel_function)(k,k);
      QD[k+l] = QD[k];
    }
    buffer[0] = new Qfloat[2*l];
    buffer[1] = new Qfloat[2*l];
    next_buffer = 0;
  }

  void swap_index(int i, int j) const override
  {
    swap(sign[i],sign[j]);
    swap(index[i],index[j]);
    swap(QD[i],QD[j]);
  }

  Qfloat *get_Q(int i, int len) const override
  {
    Qfloat *data;
    int j, real_i = index[i];
    if(cache->get_data(real_i,&data,l) < l)
    {
      for(j=0;j<l;j++)
        data[j] = (Qfloat)(this->*kernel_function)(real_i,j);
    }

    // reorder and copy
    Qfloat *buf = buffer[next_buffer];
    next_buffer = 1 - next_buffer;
    schar si = sign[i];
    for(j=0;j<len;j++)
      buf[j] = (Qfloat) si * (Qfloat) sign[j] * data[index[j]];
    return buf;
  }

  double *get_QD() const override
  {
    return QD;
  }

  ~SVR_Q() override
  {
    delete cache;
    delete[] sign;
    delete[] index;
    delete[] buffer[0];
    delete[] buffer[1];
    delete[] QD;
  }
private:
  int l;
  Cache *cache;
  schar *sign;
  int *index;
  mutable int next_buffer;
  Qfloat *buffer[2];
  double *QD;
};

//
// construct and solve various formulations
//
static void solve_c_svc(
    const svm_problem *prob, const svm_parameter* param,
    double *alpha, Solver::SolutionInfo* si, double Cp, double Cn)
{
  int l = prob->l;
  auto *minus_ones = new double[l];
  auto *y = new schar[l];
  auto *C = new double[l];

  int i;

  for(i=0;i<l;i++)
  {
    alpha[i] = 0;
    minus_ones[i] = -1;
    if(prob->y[i] > 0)
    {
      y[i] = +1;
      C[i] = prob->W[i]*Cp;
    }
    else
    {
      y[i] = -1;
      C[i] = prob->W[i]*Cn;
    }
  }

  Solver s;
  s.Solve(l, SVC_Q(*prob,*param,y), minus_ones, y,
          alpha, C, param->eps, si, param->shrinking);

  /*
 double sum_alpha=0;
 for(i=0;i<l;i++)
  sum_alpha += alpha[i];
 if (Cp==Cn)
  info("nu = %f\n", sum_alpha/(Cp*prob->l));
 */

  for(i=0;i<l;i++)
    alpha[i] *= y[i];

  delete[] C;
  delete[] minus_ones;
  delete[] y;
}

static void solve_nu_svc(
    const svm_problem *prob, const svm_parameter *param,
    double *alpha, Solver::SolutionInfo* si)
{
  int i;
  int l = prob->l;
  double nu = param->nu;

  auto *y = new schar[l];
  auto *C = new double[l];

  for(i=0;i<l;i++)
  {
    if(prob->y[i]>0)
      y[i] = +1;
    else
      y[i] = -1;
    C[i] = prob->W[i];
  }

  double nu_l = 0;
  for(i=0;i<l;i++) nu_l += nu*C[i];
  double sum_pos = nu_l/2;
  double sum_neg = nu_l/2;

  for(i=0;i<l;i++)
    if(y[i] == +1)
    {
      alpha[i] = min(C[i],sum_pos);
      sum_pos -= alpha[i];
    }
    else
    {
      alpha[i] = min(C[i],sum_neg);
      sum_neg -= alpha[i];
    }

  auto *zeros = new double[l];

  for(i=0;i<l;i++)
    zeros[i] = 0;

  Solver_NU s;
  s.Solve(l, SVC_Q(*prob,*param,y), zeros, y,
          alpha, C, param->eps, si,  param->shrinking);
  double r = si->r;

  info("C = %f\n",1/r);

  for(i=0;i<l;i++)
  {
    alpha[i] *= y[i]/r;
    si->upper_bound[i] /= r;
  }

  si->rho /= r;
  si->obj /= (r*r);

  delete[] C;
  delete[] y;
  delete[] zeros;
}

static void solve_one_class(
    const svm_problem *prob, const svm_parameter *param,
    double *alpha, Solver::SolutionInfo* si)
{
  int l = prob->l;
  auto *zeros = new double[l];
  auto *ones = new schar[l];
  auto *C = new double[l];
  int i;

  double nu_l = 0;

  for(i=0;i<l;i++)
  {
    C[i] = prob->W[i];
    nu_l += C[i] * param->nu;
  }

  i = 0;
  while(nu_l > 0)
  {
    alpha[i] = min(C[i],nu_l);
    nu_l -= alpha[i];
    ++i;
  }
  for(;i<l;i++)
    alpha[i] = 0;

  for(i=0;i<l;i++)
  {
    zeros[i] = 0;
    ones[i] = 1;
  }

  Solver s;
  s.Solve(l, ONE_CLASS_Q(*prob,*param), zeros, ones,
          alpha, C, param->eps, si, param->shrinking);

  delete[] C;
  delete[] zeros;
  delete[] ones;
}

static void solve_epsilon_svr(
    const svm_problem *prob, const svm_parameter *param,
    double *alpha, Solver::SolutionInfo* si)
{
  int l = prob->l;
  auto *alpha2 = new double[2*l];
  auto *linear_term = new double[2*l];
  auto *C = new double[2*l];
  auto *y = new schar[2*l];
  int i;

  for(i=0;i<l;i++)
  {
    alpha2[i] = 0;
    linear_term[i] = param->p - prob->y[i];
    y[i] = 1;
    C[i] = prob->W[i]*param->C;

    alpha2[i+l] = 0;
    linear_term[i+l] = param->p + prob->y[i];
    y[i+l] = -1;
    C[i+l] = prob->W[i]*param->C;
  }

  Solver s;
  s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y,
          alpha2, C, param->eps, si, param->shrinking);
  double sum_alpha = 0;
  for(i=0;i<l;i++)
  {
    alpha[i] = alpha2[i] - alpha2[i+l];
    sum_alpha += fabs(alpha[i]);
  }
  //info("nu = %f\n",sum_alpha/(param->C*l));
  delete[] alpha2;
  delete[] linear_term;
  delete[] C;
  delete[] y;
}

static void solve_nu_svr(
    const svm_problem *prob, const svm_parameter *param,
    double *alpha, Solver::SolutionInfo* si)
{
  int l = prob->l;
  auto *C = new double[2*l];
  auto *alpha2 = new double[2*l];
  auto *linear_term = new double[2*l];
  auto *y = new schar[2*l];
  int i;

  double sum = 0;
  for(i=0;i<l;i++)
  {
    C[i] = C[i+l] = prob->W[i]*param->C;
    sum += C[i] * param->nu;
  }
  sum /= 2;

  for(i=0;i<l;i++)
  {
    alpha2[i] = alpha2[i+l] = min(sum,C[i]);
    sum -= alpha2[i];

    linear_term[i] = - prob->y[i];
    y[i] = 1;

    linear_term[i+l] = prob->y[i];
    y[i+l] = -1;
  }

  Solver_NU s;
  s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y,
          alpha2, C, param->eps, si, param->shrinking);

  info("epsilon = %f\n",-si->r);

  for(i=0;i<l;i++)
    alpha[i] = alpha2[i] - alpha2[i+l];

  delete[] alpha2;
  delete[] linear_term;
  delete[] C;
  delete[] y;
}

//
// decision_function
//
struct decision_function
{
  double *alpha;
  double rho;
};

static decision_function svm_train_one(
    const svm_problem *prob, const svm_parameter *param,
    double Cp, double Cn)
{
  auto *alpha = Malloc(double,prob->l);
  Solver::SolutionInfo si;
  switch(param->svm_type)
  {
  case C_SVC:
    si.upper_bound = Malloc(double,prob->l);
    solve_c_svc(prob,param,alpha,&si,Cp,Cn);
    break;
  case NU_SVC:
    si.upper_bound = Malloc(double,prob->l);
    solve_nu_svc(prob,param,alpha,&si);
    break;
  case ONE_CLASS:
    si.upper_bound = Malloc(double,prob->l);
    solve_one_class(prob,param,alpha,&si);
    break;
  case EPSILON_SVR:
    si.upper_bound = Malloc(double,2*prob->l);
    solve_epsilon_svr(prob,param,alpha,&si);
    break;
  case NU_SVR:
    si.upper_bound = Malloc(double,2*prob->l);
    solve_nu_svr(prob,param,alpha,&si);
    break;
  }

  info("obj = %f, rho = %f\n",si.obj,si.rho);

  // output SVs

  int nSV = 0;
  int nBSV = 0;
  for(int i=0;i<prob->l;i++)
  {
    if(fabs(alpha[i]) > 0)
    {
      ++nSV;
      if(prob->y[i] > 0)
      {
        if(fabs(alpha[i]) >= si.upper_bound[i])
          ++nBSV;
      }
      else
      {
        if(fabs(alpha[i]) >= si.upper_bound[i])
          ++nBSV;
      }
    }
  }

  free(si.upper_bound);

  info("nSV = %d, nBSV = %d\n",nSV,nBSV);

  decision_function f;
  f.alpha = alpha;
  f.rho = si.rho;
  return f;
}

// Platt's binary SVM Probablistic Output: an improvement from Lin et al.
static void sigmoid_train(
    int l, const double *dec_values, const double *labels,
    double& A, double& B)
{
  double prior1=0, prior0 = 0;
  int i;

  for (i=0;i<l;i++)
    if (labels[i] > 0) prior1+=1;
    else prior0+=1;

  int max_iter=100; // Maximal number of iterations
  double min_step=1e-10; // Minimal step taken in line search
  double sigma=1e-12; // For numerically strict PD of Hessian
  double eps=1e-5;
  double hiTarget=(prior1+1.0)/(prior1+2.0);
  double loTarget=1/(prior0+2.0);
  auto *t=Malloc(double,l);
  double fApB,p,q,h11,h22,h21,g1,g2,det,dA,dB,gd,stepsize;
  double newA,newB,newf,d1,d2;
  int iter;

  // Initial Point and Initial Fun Value
  A=0.0; B=log((prior0+1.0)/(prior1+1.0));
  double fval = 0.0;

  for (i=0;i<l;i++)
  {
    if (labels[i]>0) t[i]=hiTarget;
    else t[i]=loTarget;
    fApB = dec_values[i]*A+B;
    if (fApB>=0)
      fval += t[i]*fApB + log(1+exp(-fApB));
    else
      fval += (t[i] - 1)*fApB +log(1+exp(fApB));
  }
  for (iter=0;iter<max_iter;iter++)
  {
    // Update Gradient and Hessian (use H' = H + sigma I)
    h11=sigma; // numerically ensures strict PD
    h22=sigma;
    h21=0.0;g1=0.0;g2=0.0;
    for (i=0;i<l;i++)
    {
      fApB = dec_values[i]*A+B;
      if (fApB >= 0)
      {
        p=exp(-fApB)/(1.0+exp(-fApB));
        q=1.0/(1.0+exp(-fApB));
      }
      else
      {
        p=1.0/(1.0+exp(fApB));
        q=exp(fApB)/(1.0+exp(fApB));
      }
      d2=p*q;
      h11+=dec_values[i]*dec_values[i]*d2;
      h22+=d2;
      h21+=dec_values[i]*d2;
      d1=t[i]-p;
      g1+=dec_values[i]*d1;
      g2+=d1;
    }

    // Stopping Criteria
    if (fabs(g1)<eps && fabs(g2)<eps)
      break;

    // Finding Newton direction: -inv(H') * g
    det=h11*h22-h21*h21;
    dA=-(h22*g1 - h21 * g2) / det;
    dB=-(-h21*g1+ h11 * g2) / det;
    gd=g1*dA+g2*dB;


    stepsize = 1;  // Line Search
    while (stepsize >= min_step)
    {
      newA = A + stepsize * dA;
      newB = B + stepsize * dB;

      // New function value
      newf = 0.0;
      for (i=0;i<l;i++)
      {
        fApB = dec_values[i]*newA+newB;
        if (fApB >= 0)
          newf += t[i]*fApB + log(1+exp(-fApB));
        else
          newf += (t[i] - 1)*fApB +log(1+exp(fApB));
      }
      // Check sufficient decrease
      if (newf<fval+0.0001*stepsize*gd)
      {
        A=newA;B=newB;fval=newf;
        break;
      }
      else
        stepsize = stepsize / 2.0;
    }

    if (stepsize < min_step)
    {
      info("Line search fails in two-class probability estimates\n");
      break;
    }
  }

  if (iter>=max_iter)
    info("Reaching maximal iterations in two-class probability estimates\n");
  free(t);
}

static double sigmoid_predict(double decision_value, double A, double B)
{
  double fApB = decision_value*A+B;
  // 1-p used later; avoid catastrophic cancellation
  if (fApB >= 0)
    return exp(-fApB)/(1.0+exp(-fApB));
  else
    return 1.0/(1+exp(fApB)) ;
}

// Method 2 from the multiclass_prob paper by Wu, Lin, and Weng
static void multiclass_probability(int k, double **r, double *p)
{
  int t,j;
  int iter = 0, max_iter=max(100,k);
  auto **Q=Malloc(double *,k);
  auto *Qp=Malloc(double,k);
  double pQp, eps=0.005/k;

  for (t=0;t<k;t++)
  {
    p[t]=1.0/k;  // Valid if k = 1
    Q[t]=Malloc(double,k);
    Q[t][t]=0;
    for (j=0;j<t;j++)
    {
      Q[t][t]+=r[j][t]*r[j][t];
      Q[t][j]=Q[j][t];
    }
    for (j=t+1;j<k;j++)
    {
      Q[t][t]+=r[j][t]*r[j][t];
      Q[t][j]=-r[j][t]*r[t][j];
    }
  }
  for (iter=0;iter<max_iter;iter++)
  {
    // stopping condition, recalculate QP,pQP for numerical accuracy
    pQp=0;
    for (t=0;t<k;t++)
    {
      Qp[t]=0;
      for (j=0;j<k;j++)
        Qp[t]+=Q[t][j]*p[j];
      pQp+=p[t]*Qp[t];
    }
    double max_error=0;
    for (t=0;t<k;t++)
    {
      double error=fabs(Qp[t]-pQp);
      if (error>max_error)
        max_error=error;
    }
    if (max_error<eps) break;

    for (t=0;t<k;t++)
    {
      double diff=(-Qp[t]+pQp)/Q[t][t];
      p[t]+=diff;
      pQp=(pQp+diff*(diff*Q[t][t]+2*Qp[t]))/(1+diff)/(1+diff);
      for (j=0;j<k;j++)
      {
        Qp[j]=(Qp[j]+diff*Q[t][j])/(1+diff);
        p[j]/=(1+diff);
      }
    }
  }
  if (iter>=max_iter)
    info("Exceeds max_iter in multiclass_prob\n");
  for(t=0;t<k;t++) free(Q[t]);
  free(Q);
  free(Qp);
}

// Cross-validation decision values for probability estimates
static void svm_binary_svc_probability(
    const svm_problem *prob, const svm_parameter *param,
    double Cp, double Cn, double& probA, double& probB)
{
  int i;
  int nr_fold = 5;
  auto *perm = Malloc(int,prob->l);
  auto *dec_values = Malloc(double,prob->l);

  // random shuffle
  for(i=0;i<prob->l;i++) perm[i]=i;
  for(i=0;i<prob->l;i++)
  {
    int j = i+rand()%(prob->l-i);
    swap(perm[i],perm[j]);
  }
  for(i=0;i<nr_fold;i++)
  {
    int begin = i*prob->l/nr_fold;
    int end = (i+1)*prob->l/nr_fold;
    int j,k;
    struct svm_problem subprob;

    subprob.l = prob->l-(end-begin);
    subprob.x = Malloc(struct svm_node*,subprob.l);
    subprob.y = Malloc(double,subprob.l);
    subprob.W = Malloc(double,subprob.l);

    k=0;
    for(j=0;j<begin;j++)
    {
      subprob.x[k] = prob->x[perm[j]];
      subprob.y[k] = prob->y[perm[j]];
      subprob.W[k] = prob->W[perm[j]];
      ++k;
    }
    for(j=end;j<prob->l;j++)
    {
      subprob.x[k] = prob->x[perm[j]];
      subprob.y[k] = prob->y[perm[j]];
      subprob.W[k] = prob->W[perm[j]];
      ++k;
    }
    int p_count=0,n_count=0;
    for(j=0;j<k;j++)
      if(subprob.y[j]>0)
        p_count++;
      else
        n_count++;

    if(p_count==0 && n_count==0)
      for(j=begin;j<end;j++)
        dec_values[perm[j]] = 0;
    else if(p_count > 0 && n_count == 0)
      for(j=begin;j<end;j++)
        dec_values[perm[j]] = 1;
    else if(p_count == 0 && n_count > 0)
      for(j=begin;j<end;j++)
        dec_values[perm[j]] = -1;
    else
    {
      svm_parameter subparam = *param;
      subparam.probability=0;
      subparam.C=1.0;
      subparam.nr_weight=2;
      subparam.weight_label = Malloc(int,2);
      subparam.weight = Malloc(double,2);
      subparam.weight_label[0]=+1;
      subparam.weight_label[1]=-1;
      subparam.weight[0]=Cp;
      subparam.weight[1]=Cn;
      struct svm_model *submodel = svm_train(&subprob,&subparam);
      for(j=begin;j<end;j++)
      {
        svm_predict_values(submodel,prob->x[perm[j]],&(dec_values[perm[j]]));
        // ensure +1 -1 order; reason not using CV subroutine
        dec_values[perm[j]] *= submodel->label[0];
      }
      svm_free_and_destroy_model(&submodel);
      svm_destroy_param(&subparam);
    }
    free(subprob.x);
    free(subprob.y);
    free(subprob.W);
  }
  sigmoid_train(prob->l,dec_values,prob->y,probA,probB);
  free(dec_values);
  free(perm);
}

// Return parameter of a Laplace distribution
static double svm_svr_probability(
    const svm_problem *prob, const svm_parameter *param)
{
  int i;
  int nr_fold = 5;
  auto *ymv = Malloc(double,prob->l);
  double mae = 0;

  svm_parameter newparam = *param;
  newparam.probability = 0;
  svm_cross_validation(prob,&newparam,nr_fold,ymv);
  for(i=0;i<prob->l;i++)
  {
    ymv[i]=prob->y[i]-ymv[i];
    mae += fabs(ymv[i]);
  }
  mae /= prob->l;
  double std=sqrt(2*mae*mae);
  int count=0;
  mae=0;
  for(i=0;i<prob->l;i++)
    if (fabs(ymv[i]) > 5*std)
      count=count+1;
    else
      mae+=fabs(ymv[i]);
  mae /= (prob->l-count);
  info("Prob. model for test data: target value = predicted value + z,\nz: Laplace distribution e^(-|z|/sigma)/(2sigma),sigma= %g\n",mae);
  free(ymv);
  return mae;
}


// label: label name, start: begin of each class, count: #data of classes, perm: indices to the original data
// perm, length l, must be allocated before calling this subroutine
static void svm_group_classes(const svm_problem *prob, int *nr_class_ret, int **label_ret, int **start_ret, int **count_ret, int *perm)
{
  int l = prob->l;
  int max_nr_class = 16;
  int nr_class = 0;
  auto *label = Malloc(int,max_nr_class);
  auto *count = Malloc(int,max_nr_class);
  auto *data_label = Malloc(int,l);
  int i;

  for(i=0;i<l;i++)
  {
    auto this_label = (int)prob->y[i];
    int j;
    for(j=0;j<nr_class;j++)
    {
      if(this_label == label[j])
      {
        ++count[j];
        break;
      }
    }
    data_label[i] = j;
    if(j == nr_class)
    {
      if(nr_class == max_nr_class)
      {
        max_nr_class *= 2;
        label = (int *)realloc(label,max_nr_class*sizeof(int));
        count = (int *)realloc(count,max_nr_class*sizeof(int));
      }
      label[nr_class] = this_label;
      count[nr_class] = 1;
      ++nr_class;
    }
  }

  //
  // Labels are ordered by their first occurrence in the training set.
  // However, for two-class sets with -1/+1 labels and -1 appears first,
  // we swap labels to ensure that internally the binary SVM has positive data corresponding to the +1 instances.
  //
  if (nr_class == 2 && label[0] == -1 && label[1] == 1)
  {
    swap(label[0],label[1]);
    swap(count[0],count[1]);
    for(i=0;i<l;i++)
    {
      if(data_label[i] == 0)
        data_label[i] = 1;
      else
        data_label[i] = 0;
    }
  }

  auto *start = Malloc(int,nr_class);
  start[0] = 0;
  for(i=1;i<nr_class;i++)
    start[i] = start[i-1]+count[i-1];
  for(i=0;i<l;i++)
  {
    perm[start[data_label[i]]] = i;
    ++start[data_label[i]];
  }
  start[0] = 0;
  for(i=1;i<nr_class;i++)
    start[i] = start[i-1]+count[i-1];

  *nr_class_ret = nr_class;
  *label_ret = label;
  *start_ret = start;
  *count_ret = count;
  free(data_label);
}

//
// Remove zero weighed data as libsvm and some liblinear solvers require C > 0.
//
static void remove_zero_weight(svm_problem *newprob, const svm_problem *prob)
{
  int i;
  int l = 0;
  for(i=0;i<prob->l;i++)
    if(prob->W[i] > 0) l++;
  *newprob = *prob;
  newprob->l = l;
  newprob->x = Malloc(svm_node*,l);
  newprob->y = Malloc(double,l);
  newprob->W = Malloc(double,l);

  int j = 0;
  for(i=0;i<prob->l;i++)
    if(prob->W[i] > 0)
    {
      newprob->x[j] = prob->x[i];
      newprob->y[j] = prob->y[i];
      newprob->W[j] = prob->W[i];
      j++;
    }
}

//
// Interface functions
//
svm_model *svm_train(const svm_problem *prob, const svm_parameter *param)
{
  svm_problem newprob;
  remove_zero_weight(&newprob, prob);
  prob = &newprob;

  auto *model = Malloc(svm_model,1);
  model->param = *param;
  model->free_sv = 0; // XXX

  if(param->svm_type == ONE_CLASS ||
     param->svm_type == EPSILON_SVR ||
     param->svm_type == NU_SVR)
  {
    // regression or one-class-svm
    model->nr_class = 2;
    model->label = nullptr;
    model->nSV = nullptr;
    model->probA = nullptr; model->probB = nullptr;
    model->sv_coef = Malloc(double *,1);

    if(param->probability &&
       (param->svm_type == EPSILON_SVR ||
        param->svm_type == NU_SVR))
    {
      model->probA = Malloc(double,1);
      model->probA[0] = svm_svr_probability(prob,param);
    }

    decision_function f = svm_train_one(prob,param,0,0);
    model->rho = Malloc(double,1);
    model->rho[0] = f.rho;

    int nSV = 0;
    int i;
    for(i=0;i<prob->l;i++)
      if(fabs(f.alpha[i]) > 0) ++nSV;
    model->l = nSV;
    model->SV = Malloc(svm_node *,nSV);
    model->sv_coef[0] = Malloc(double,nSV);
    model->sv_indices = Malloc(int,nSV);
    int j = 0;
    for(i=0;i<prob->l;i++)
      if(fabs(f.alpha[i]) > 0)
      {
        model->SV[j] = prob->x[i];
        model->sv_coef[0][j] = f.alpha[i];
        model->sv_indices[j] = i+1;
        ++j;
      }

    free(f.alpha);
  }
  else
  {
    // classification
    int l = prob->l;
    int nr_class;
    int *label = nullptr;
    int *start = nullptr;
    int *count = nullptr;
    auto *perm = Malloc(int,l);

    // group training data of the same class
    svm_group_classes(prob,&nr_class,&label,&start,&count,perm);
    if(nr_class == 1)
      info("WARNING: training data in only one class. See README for details.\n");

    auto **x = Malloc(svm_node *,l);
    double *W;
    W = Malloc(double,l);

    int i;
    for(i=0;i<l;i++)
    {
      x[i] = prob->x[perm[i]];
      W[i] = prob->W[perm[i]];
    }

    // calculate weighted C

    auto *weighted_C = Malloc(double, nr_class);
    for(i=0;i<nr_class;i++)
      weighted_C[i] = param->C;
    for(i=0;i<param->nr_weight;i++)
    {
      int j;
      for(j=0;j<nr_class;j++)
        if(param->weight_label[i] == label[j])
          break;
      if(j == nr_class)
        fprintf(stderr,"WARNING: class label %d specified in weight is not found\n", param->weight_label[i]);
      else
        weighted_C[j] *= param->weight[i];
    }

    // train k*(k-1)/2 models

    auto *nonzero = Malloc(bool,l);
    for(i=0;i<l;i++)
      nonzero[i] = false;
    auto *f = Malloc(decision_function,nr_class*(nr_class-1)/2);

    double *probA=nullptr,*probB=nullptr;
    if (param->probability)
    {
      probA=Malloc(double,nr_class*(nr_class-1)/2);
      probB=Malloc(double,nr_class*(nr_class-1)/2);
    }

    int p = 0;
    for(i=0;i<nr_class;i++)
      for(int j=i+1;j<nr_class;j++)
      {
        svm_problem sub_prob;
        int si = start[i], sj = start[j];
        int ci = count[i], cj = count[j];
        sub_prob.l = ci+cj;
        sub_prob.x = Malloc(svm_node *,sub_prob.l);
        sub_prob.y = Malloc(double,sub_prob.l);
        sub_prob.W = Malloc(double,sub_prob.l);
        int k;
        for(k=0;k<ci;k++)
        {
          sub_prob.x[k] = x[si+k];
          sub_prob.y[k] = +1;
          sub_prob.W[k] = W[si+k];
        }
        for(k=0;k<cj;k++)
        {
          sub_prob.x[ci+k] = x[sj+k];
          sub_prob.y[ci+k] = -1;
          sub_prob.W[ci+k] = W[sj+k];
        }

        if(param->probability)
          svm_binary_svc_probability(&sub_prob,param,weighted_C[i],weighted_C[j],probA[p],probB[p]);

        f[p] = svm_train_one(&sub_prob,param,weighted_C[i],weighted_C[j]);
        for(k=0;k<ci;k++)
          if(!nonzero[si+k] && fabs(f[p].alpha[k]) > 0)
            nonzero[si+k] = true;
        for(k=0;k<cj;k++)
          if(!nonzero[sj+k] && fabs(f[p].alpha[ci+k]) > 0)
            nonzero[sj+k] = true;
        free(sub_prob.x);
        free(sub_prob.y);
        free(sub_prob.W);
        ++p;
      }

    // build output

    model->nr_class = nr_class;

    model->label = Malloc(int,nr_class);
    for(i=0;i<nr_class;i++)
      model->label[i] = label[i];

    model->rho = Malloc(double,nr_class*(nr_class-1)/2);
    for(i=0;i<nr_class*(nr_class-1)/2;i++)
      model->rho[i] = f[i].rho;

    if(param->probability)
    {
      model->probA = Malloc(double,nr_class*(nr_class-1)/2);
      model->probB = Malloc(double,nr_class*(nr_class-1)/2);
      for(i=0;i<nr_class*(nr_class-1)/2;i++)
      {
        model->probA[i] = probA[i];
        model->probB[i] = probB[i];
      }
    }
    else
    {
      model->probA=nullptr;
      model->probB=nullptr;
    }

    int total_sv = 0;
    auto *nz_count = Malloc(int,nr_class);
    model->nSV = Malloc(int,nr_class);
    for(i=0;i<nr_class;i++)
    {
      int nSV = 0;
      for(int j=0;j<count[i];j++)
        if(nonzero[start[i]+j])
        {
          ++nSV;
          ++total_sv;
        }
      model->nSV[i] = nSV;
      nz_count[i] = nSV;
    }

    info("Total nSV = %d\n",total_sv);

    model->l = total_sv;
    model->SV = Malloc(svm_node *,total_sv);
    model->sv_indices = Malloc(int,total_sv);
    p = 0;
    for(i=0;i<l;i++)
      if(nonzero[i])
      {
        model->SV[p] = x[i];
        model->sv_indices[p++] = perm[i] + 1;
      }

    auto *nz_start = Malloc(int,nr_class);
    nz_start[0] = 0;
    for(i=1;i<nr_class;i++)
      nz_start[i] = nz_start[i-1]+nz_count[i-1];

    model->sv_coef = Malloc(double *,nr_class-1);
    for(i=0;i<nr_class-1;i++)
      model->sv_coef[i] = Malloc(double,total_sv);

    p = 0;
    for(i=0;i<nr_class;i++)
      for(int j=i+1;j<nr_class;j++)
      {
        // classifier (i,j): coefficients with
        // i are in sv_coef[j-1][nz_start[i]...],
        // j are in sv_coef[i][nz_start[j]...]

        int si = start[i];
        int sj = start[j];
        int ci = count[i];
        int cj = count[j];

        int q = nz_start[i];
        int k;
        for(k=0;k<ci;k++)
          if(nonzero[si+k])
            model->sv_coef[j-1][q++] = f[p].alpha[k];
        q = nz_start[j];
        for(k=0;k<cj;k++)
          if(nonzero[sj+k])
            model->sv_coef[i][q++] = f[p].alpha[ci+k];
        ++p;
      }

    free(label);
    free(probA);
    free(probB);
    free(count);
    free(perm);
    free(start);
    free(W);
    free(x);
    free(weighted_C);
    free(nonzero);
    for(i=0;i<nr_class*(nr_class-1)/2;i++)
      free(f[i].alpha);
    free(f);
    free(nz_count);
    free(nz_start);
  }
  free(newprob.x);
  free(newprob.y);
  free(newprob.W);
  return model;
}

// Stratified cross validation
void svm_cross_validation(const svm_problem *prob, const svm_parameter *param, int nr_fold, double *target)
{
  int i;
  int *fold_start;
  int l = prob->l;
  auto *perm = Malloc(int,l);
  int nr_class;
  if (nr_fold > l)
  {
    nr_fold = l;
    fprintf(stderr,"WARNING: # folds > # data. Will use # folds = # data instead (i.e., leave-one-out cross validation)\n");
  }
  fold_start = Malloc(int,nr_fold+1);
  // stratified cv may not give leave-one-out rate
  // Each class to l folds -> some folds may have zero elements
  if((param->svm_type == C_SVC ||
      param->svm_type == NU_SVC) && nr_fold < l)
  {
    int *start = nullptr;
    int *label = nullptr;
    int *count = nullptr;
    svm_group_classes(prob,&nr_class,&label,&start,&count,perm);

    // random shuffle and then data grouped by fold using the array perm
    auto *fold_count = Malloc(int,nr_fold);
    int c;
    auto *index = Malloc(int,l);
    for(i=0;i<l;i++)
      index[i]=perm[i];
    for (c=0; c<nr_class; c++)
      for(i=0;i<count[c];i++)
      {
        int j = i+rand()%(count[c]-i);
        swap(index[start[c]+j],index[start[c]+i]);
      }
    for(i=0;i<nr_fold;i++)
    {
      fold_count[i] = 0;
      for (c=0; c<nr_class;c++)
        fold_count[i]+=(i+1)*count[c]/nr_fold-i*count[c]/nr_fold;
    }
    fold_start[0]=0;
    for (i=1;i<=nr_fold;i++)
      fold_start[i] = fold_start[i-1]+fold_count[i-1];
    for (c=0; c<nr_class;c++)
      for(i=0;i<nr_fold;i++)
      {
        int begin = start[c]+i*count[c]/nr_fold;
        int end = start[c]+(i+1)*count[c]/nr_fold;
        for(int j=begin;j<end;j++)
        {
          perm[fold_start[i]] = index[j];
          fold_start[i]++;
        }
      }
    fold_start[0]=0;
    for (i=1;i<=nr_fold;i++)
      fold_start[i] = fold_start[i-1]+fold_count[i-1];
    free(start);
    free(label);
    free(count);
    free(index);
    free(fold_count);
  }
  else
  {
    for(i=0;i<l;i++) perm[i]=i;
    for(i=0;i<l;i++)
    {
      int j = i+rand()%(l-i);
      swap(perm[i],perm[j]);
    }
    for(i=0;i<=nr_fold;i++)
      fold_start[i]=i*l/nr_fold;
  }

  for(i=0;i<nr_fold;i++)
  {
    int begin = fold_start[i];
    int end = fold_start[i+1];
    int j,k;
    struct svm_problem subprob;

    subprob.l = l-(end-begin);
    subprob.x = Malloc(struct svm_node*,subprob.l);
    subprob.y = Malloc(double,subprob.l);

    subprob.W = Malloc(double,subprob.l);
    k=0;
    for(j=0;j<begin;j++)
    {
      subprob.x[k] = prob->x[perm[j]];
      subprob.y[k] = prob->y[perm[j]];
      subprob.W[k] = prob->W[perm[j]];
      ++k;
    }
    for(j=end;j<l;j++)
    {
      subprob.x[k] = prob->x[perm[j]];
      subprob.y[k] = prob->y[perm[j]];
      subprob.W[k] = prob->W[perm[j]];
      ++k;
    }
    struct svm_model *submodel = svm_train(&subprob,param);
    if(param->probability &&
       (param->svm_type == C_SVC || param->svm_type == NU_SVC))
    {
      auto *prob_estimates=Malloc(double,svm_get_nr_class(submodel));
      for(j=begin;j<end;j++)
        target[perm[j]] = svm_predict_probability(submodel,prob->x[perm[j]],prob_estimates);
      free(prob_estimates);
    }
    else
      for(j=begin;j<end;j++)
        target[perm[j]] = svm_predict(submodel,prob->x[perm[j]]);
    svm_free_and_destroy_model(&submodel);
    free(subprob.x);
    free(subprob.y);
    free(subprob.W);
  }
  free(fold_start);
  free(perm);
}


int svm_get_svm_type(const svm_model *model)
{
  return model->param.svm_type;
}

int svm_get_nr_class(const svm_model *model)
{
  return model->nr_class;
}

void svm_get_labels(const svm_model *model, int* label)
{
  if (model->label != nullptr)
    for(int i=0;i<model->nr_class;i++)
      label[i] = model->label[i];
}

void svm_get_sv_indices(const svm_model *model, int* indices)
{
  if (model->sv_indices != nullptr)
    for(int i=0;i<model->l;i++)
      indices[i] = model->sv_indices[i];
}

int svm_get_nr_sv(const svm_model *model)
{
  return model->l;
}

double svm_get_svr_probability(const svm_model *model)
{
  if ((model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) &&
      model->probA!=nullptr)
    return model->probA[0];
  else
  {
    fprintf(stderr,"Model doesn't contain information for SVR probability inference\n");
    return 0;
  }
}

double svm_predict_values(const svm_model *model, const svm_node *x, double* dec_values)
{
  int i;
  if(model->param.svm_type == ONE_CLASS ||
     model->param.svm_type == EPSILON_SVR ||
     model->param.svm_type == NU_SVR)
  {
    double *sv_coef = model->sv_coef[0];
    double sum = 0;
    for(i=0;i<model->l;i++)
      sum += sv_coef[i] * Kernel::k_function(x,model->SV[i],model->param);
    sum -= model->rho[0];
    *dec_values = sum;

    if(model->param.svm_type == ONE_CLASS)
      return (sum>0)?1:-1;
    else
      return sum;
  }
  else
  {
    int nr_class = model->nr_class;
    int l = model->l;

    auto *kvalue = Malloc(double,l);
    for(i=0;i<l;i++)
      kvalue[i] = Kernel::k_function(x,model->SV[i],model->param);

    auto *start = Malloc(int,nr_class);
    start[0] = 0;
    for(i=1;i<nr_class;i++)
      start[i] = start[i-1]+model->nSV[i-1];

    auto *vote = Malloc(int,nr_class);
    for(i=0;i<nr_class;i++)
      vote[i] = 0;

    int p=0;
    for(i=0;i<nr_class;i++)
      for(int j=i+1;j<nr_class;j++)
      {
        double sum = 0;
        int si = start[i];
        int sj = start[j];
        int ci = model->nSV[i];
        int cj = model->nSV[j];

        int k;
        double *coef1 = model->sv_coef[j-1];
        double *coef2 = model->sv_coef[i];
        for(k=0;k<ci;k++)
          sum += coef1[si+k] * kvalue[si+k];
        for(k=0;k<cj;k++)
          sum += coef2[sj+k] * kvalue[sj+k];
        sum -= model->rho[p];
        dec_values[p] = sum;

        if(dec_values[p] > 0)
          ++vote[i];
        else
          ++vote[j];
        p++;
      }

    int vote_max_idx = 0;
    for(i=1;i<nr_class;i++)
      if(vote[i] > vote[vote_max_idx])
        vote_max_idx = i;

    free(kvalue);
    free(start);
    free(vote);
    return model->label[vote_max_idx];
  }
}

double svm_predict(const svm_model *model, const svm_node *x)
{
  int nr_class = model->nr_class;
  double *dec_values;
  if(model->param.svm_type == ONE_CLASS ||
     model->param.svm_type == EPSILON_SVR ||
     model->param.svm_type == NU_SVR)
    dec_values = Malloc(double, 1);
  else
    dec_values = Malloc(double, nr_class*(nr_class-1)/2);
  double pred_result = svm_predict_values(model, x, dec_values);
  free(dec_values);
  return pred_result;
}

double svm_predict_probability(
    const svm_model *model, const svm_node *x, double *prob_estimates)
{
  if ((model->param.svm_type == C_SVC || model->param.svm_type == NU_SVC) &&
      model->probA!=nullptr && model->probB!=nullptr)
  {
    int i;
    int nr_class = model->nr_class;
    auto *dec_values = Malloc(double, nr_class*(nr_class-1)/2);
    svm_predict_values(model, x, dec_values);

    double min_prob=1e-7;
    auto **pairwise_prob=Malloc(double *,nr_class);
    for(i=0;i<nr_class;i++)
      pairwise_prob[i]=Malloc(double,nr_class);
    int k=0;
    for(i=0;i<nr_class;i++)
      for(int j=i+1;j<nr_class;j++)
      {
        pairwise_prob[i][j]=min(max(sigmoid_predict(dec_values[k],model->probA[k],model->probB[k]),min_prob),1-min_prob);
        pairwise_prob[j][i]=1-pairwise_prob[i][j];
        k++;
      }
    multiclass_probability(nr_class,pairwise_prob,prob_estimates);

    int prob_max_idx = 0;
    for(i=1;i<nr_class;i++)
      if(prob_estimates[i] > prob_estimates[prob_max_idx])
        prob_max_idx = i;
    for(i=0;i<nr_class;i++)
      free(pairwise_prob[i]);
    free(dec_values);
    free(pairwise_prob);
    return model->label[prob_max_idx];
  }
  else
    return svm_predict(model, x);
}

static const char *svm_type_table[] =
{
  "c_svc","nu_svc","one_class","epsilon_svr","nu_svr",nullptr
};

static const char *kernel_type_table[]=
{
  "linear","polynomial","rbf","sigmoid","precomputed",nullptr
};

int svm_save_model(const char *model_file_name, const svm_model *model)
{
  FILE *fp = fopen(model_file_name,"w");
  if(fp==nullptr) return -1;

  mitk::LocaleSwitch localeSwitch("C");

  const svm_parameter& param = model->param;

  fprintf(fp,"svm_type %s\n", svm_type_table[param.svm_type]);
  fprintf(fp,"kernel_type %s\n", kernel_type_table[param.kernel_type]);

  if(param.kernel_type == POLY)
    fprintf(fp,"degree %d\n", param.degree);

  if(param.kernel_type == POLY || param.kernel_type == RBF || param.kernel_type == SIGMOID)
    fprintf(fp,"gamma %g\n", param.gamma);

  if(param.kernel_type == POLY || param.kernel_type == SIGMOID)
    fprintf(fp,"coef0 %g\n", param.coef0);

  int nr_class = model->nr_class;
  int l = model->l;
  fprintf(fp, "nr_class %d\n", nr_class);
  fprintf(fp, "total_sv %d\n",l);

  {
    fprintf(fp, "rho");
    for(int i=0;i<nr_class*(nr_class-1)/2;i++)
      fprintf(fp," %g",model->rho[i]);
    fprintf(fp, "\n");
  }

  if(model->label)
  {
    fprintf(fp, "label");
    for(int i=0;i<nr_class;i++)
      fprintf(fp," %d",model->label[i]);
    fprintf(fp, "\n");
  }

  if(model->probA) // regression has probA only
  {
    fprintf(fp, "probA");
    for(int i=0;i<nr_class*(nr_class-1)/2;i++)
      fprintf(fp," %g",model->probA[i]);
    fprintf(fp, "\n");
  }
  if(model->probB)
  {
    fprintf(fp, "probB");
    for(int i=0;i<nr_class*(nr_class-1)/2;i++)
      fprintf(fp," %g",model->probB[i]);
    fprintf(fp, "\n");
  }

  if(model->nSV)
  {
    fprintf(fp, "nr_sv");
    for(int i=0;i<nr_class;i++)
      fprintf(fp," %d",model->nSV[i]);
    fprintf(fp, "\n");
  }

  fprintf(fp, "SV\n");
  const double * const *sv_coef = model->sv_coef;
  const svm_node * const *SV = model->SV;

  for(int i=0;i<l;i++)
  {
    for(int j=0;j<nr_class-1;j++)
      fprintf(fp, "%.16g ",sv_coef[j][i]);

    const svm_node *p = SV[i];

    if(param.kernel_type == PRECOMPUTED)
      fprintf(fp,"0:%d ",(int)(p->value));
    else
      while(p->index != -1)
      {
        fprintf(fp,"%d:%.8g ",p->index,p->value);
        p++;
      }
    fprintf(fp, "\n");
  }

  if (ferror(fp) != 0 || fclose(fp) != 0) return -1;
  else return 0;
}

static char *line = nullptr;
static int max_line_len;

static char* readline(FILE *input)
{
  int len;

  if(fgets(line,max_line_len,input) == nullptr)
    return nullptr;

  while(strrchr(line,'\n') == nullptr)
  {
    max_line_len *= 2;
    line = (char *) realloc(line,max_line_len);
    len = (int) strlen(line);
    if(fgets(line+len,max_line_len-len,input) == nullptr)
      break;
  }
  return line;
}

//
// FSCANF helps to handle fscanf failures.
// Its do-while block avoids the ambiguity when
// if (...)
//    FSCANF();
// is used
//
#define FSCANF(_stream, _format, _var) do{ if (fscanf(_stream, _format, _var) != 1) return false; }while(0)
bool read_model_header(FILE *fp, svm_model* model)
{
  svm_parameter& param = model->param;
  char cmd[81];
  while(true)
  {
    FSCANF(fp,"%80s",cmd);

    if(strcmp(cmd,"svm_type")==0)
    {
      FSCANF(fp,"%80s",cmd);
      int i;
      for(i=0;svm_type_table[i];i++)
      {
        if(strcmp(svm_type_table[i],cmd)==0)
        {
          param.svm_type=i;
          break;
        }
      }
      if(svm_type_table[i] == nullptr)
      {
        fprintf(stderr,"unknown svm type.\n");
        return false;
      }
    }
    else if(strcmp(cmd,"kernel_type")==0)
    {
      FSCANF(fp,"%80s",cmd);
      int i;
      for(i=0;kernel_type_table[i];i++)
      {
        if(strcmp(kernel_type_table[i],cmd)==0)
        {
          param.kernel_type=i;
          break;
        }
      }
      if(kernel_type_table[i] == nullptr)
      {
        fprintf(stderr,"unknown kernel function.\n");
        return false;
      }
    }
    else if(strcmp(cmd,"degree")==0)
      FSCANF(fp,"%d",&param.degree);
    else if(strcmp(cmd,"gamma")==0)
      FSCANF(fp,"%lf",&param.gamma);
    else if(strcmp(cmd,"coef0")==0)
      FSCANF(fp,"%lf",&param.coef0);
    else if(strcmp(cmd,"nr_class")==0)
      FSCANF(fp,"%d",&model->nr_class);
    else if(strcmp(cmd,"total_sv")==0)
      FSCANF(fp,"%d",&model->l);
    else if(strcmp(cmd,"rho")==0)
    {
      int n = model->nr_class * (model->nr_class-1)/2;
      model->rho = Malloc(double,n);
      for(int i=0;i<n;i++)
        FSCANF(fp,"%lf",&model->rho[i]);
    }
    else if(strcmp(cmd,"label")==0)
    {
      int n = model->nr_class;
      model->label = Malloc(int,n);
      for(int i=0;i<n;i++)
        FSCANF(fp,"%d",&model->label[i]);
    }
    else if(strcmp(cmd,"probA")==0)
    {
      int n = model->nr_class * (model->nr_class-1)/2;
      model->probA = Malloc(double,n);
      for(int i=0;i<n;i++)
        FSCANF(fp,"%lf",&model->probA[i]);
    }
    else if(strcmp(cmd,"probB")==0)
    {
      int n = model->nr_class * (model->nr_class-1)/2;
      model->probB = Malloc(double,n);
      for(int i=0;i<n;i++)
        FSCANF(fp,"%lf",&model->probB[i]);
    }
    else if(strcmp(cmd,"nr_sv")==0)
    {
      int n = model->nr_class;
      model->nSV = Malloc(int,n);
      for(int i=0;i<n;i++)
        FSCANF(fp,"%d",&model->nSV[i]);
    }
    else if(strcmp(cmd,"SV")==0)
    {
      while(true)
      {
        int c = getc(fp);
        if(c==EOF || c=='\n') break;
      }
      break;
    }
    else
    {
      fprintf(stderr,"unknown text in model file: [%s]\n",cmd);
      return false;
    }
  }

  return true;

}

svm_model *svm_load_model(const char *model_file_name)
{
  FILE *fp = fopen(model_file_name,"rb");
  if(fp==nullptr) return nullptr;

  mitk::LocaleSwitch localeSwitch("C");

  // read parameters

  auto *model = Malloc(svm_model,1);
  model->rho = nullptr;
  model->probA = nullptr;
  model->probB = nullptr;
  model->sv_indices = nullptr;
  model->label = nullptr;
  model->nSV = nullptr;

  // read header
  if (!read_model_header(fp, model))
  {
    fprintf(stderr, "ERROR: fscanf failed to read model\n");
    free(model->rho);
    free(model->label);
    free(model->nSV);
    free(model);
    return nullptr;
  }

  // read sv_coef and SV

  int elements = 0;
  long pos = ftell(fp);

  max_line_len = 1024;
  line = Malloc(char,max_line_len);
  char *p,*endptr,*idx,*val;

  while(readline(fp)!=nullptr)
  {
    p = strtok(line,":");
    while(true)
    {
      p = strtok(nullptr,":");
      if(p == nullptr)
        break;
      ++elements;
    }
  }
  elements += model->l;

  fseek(fp,pos,SEEK_SET);

  int m = model->nr_class - 1;
  int l = model->l;
  model->sv_coef = Malloc(double *,m);
  int i;
  for(i=0;i<m;i++)
    model->sv_coef[i] = Malloc(double,l);
  model->SV = Malloc(svm_node*,l);
  svm_node *x_space = nullptr;
  if(l>0) x_space = Malloc(svm_node,elements);

  int j=0;
  for(i=0;i<l;i++)
  {
    readline(fp);
    model->SV[i] = &x_space[j];

    p = strtok(line, " \t");
    model->sv_coef[0][i] = strtod(p,&endptr);
    for(int k=1;k<m;k++)
    {
      p = strtok(nullptr, " \t");
      model->sv_coef[k][i] = strtod(p,&endptr);
    }

    while(true)
    {
      idx = strtok(nullptr, ":");
      val = strtok(nullptr, " \t");

      if(val == nullptr)
        break;
      x_space[j].index = (int) strtol(idx,&endptr,10);
      x_space[j].value = strtod(val,&endptr);

      ++j;
    }
    x_space[j++].index = -1;
  }
  free(line);

  if (ferror(fp) != 0 || fclose(fp) != 0)
    return nullptr;

  model->free_sv = 1; // XXX
  return model;
}

void svm_free_model_content(svm_model* model_ptr)
{
  if(model_ptr->free_sv && model_ptr->l > 0 && model_ptr->SV != nullptr)
    free((void *)(model_ptr->SV[0]));
  if(model_ptr->sv_coef)
  {
    for(int i=0;i<model_ptr->nr_class-1;i++)
      free(model_ptr->sv_coef[i]);
  }

  free(model_ptr->SV);
  model_ptr->SV = nullptr;

  free(model_ptr->sv_coef);
  model_ptr->sv_coef = nullptr;

  free(model_ptr->rho);
  model_ptr->rho = nullptr;

  free(model_ptr->label);
  model_ptr->label= nullptr;

  free(model_ptr->probA);
  model_ptr->probA = nullptr;

  free(model_ptr->probB);
  model_ptr->probB= nullptr;

  free(model_ptr->sv_indices);
  model_ptr->sv_indices = nullptr;

  free(model_ptr->nSV);
  model_ptr->nSV = nullptr;
}

void svm_free_and_destroy_model(svm_model** model_ptr_ptr)
{
  if(model_ptr_ptr != nullptr && *model_ptr_ptr != nullptr)
  {
    svm_free_model_content(*model_ptr_ptr);
    free(*model_ptr_ptr);
    *model_ptr_ptr = nullptr;
  }
}

void svm_destroy_param(svm_parameter* param)
{
  free(param->weight_label);
  free(param->weight);
}

const char *svm_check_parameter(const svm_problem *prob, const svm_parameter *param)
{
  // svm_type

  int svm_type = param->svm_type;
  if(svm_type != C_SVC &&
     svm_type != NU_SVC &&
     svm_type != ONE_CLASS &&
     svm_type != EPSILON_SVR &&
     svm_type != NU_SVR)
    return "unknown svm type";

  // kernel_type, degree

  int kernel_type = param->kernel_type;
  if(kernel_type != LINEAR &&
     kernel_type != POLY &&
     kernel_type != RBF &&
     kernel_type != SIGMOID &&
     kernel_type != PRECOMPUTED)
    return "unknown kernel type";

  if(param->gamma < 0)
    return "gamma < 0";

  if(param->degree < 0)
    return "degree of polynomial kernel < 0";

  // cache_size,eps,C,nu,p,shrinking

  if(param->cache_size <= 0)
    return "cache_size <= 0";

  if(param->eps <= 0)
    return "eps <= 0";

  if(svm_type == C_SVC ||
     svm_type == EPSILON_SVR ||
     svm_type == NU_SVR)
    if(param->C <= 0)
      return "C <= 0";

  if(svm_type == NU_SVC ||
     svm_type == ONE_CLASS ||
     svm_type == NU_SVR)
    if(param->nu <= 0 || param->nu > 1)
      return "nu <= 0 or nu > 1";

  if(svm_type == EPSILON_SVR)
    if(param->p < 0)
      return "p < 0";

  if(param->shrinking != 0 &&
     param->shrinking != 1)
    return "shrinking != 0 and shrinking != 1";

  if(param->probability != 0 &&
     param->probability != 1)
    return "probability != 0 and probability != 1";

  if(param->probability == 1 &&
     svm_type == ONE_CLASS)
    return "one-class SVM probability output not supported yet";


  // check whether nu-svc is feasible

  if(svm_type == NU_SVC)
  {
    int l = prob->l;
    int max_nr_class = 16;
    int nr_class = 0;
    auto *label = Malloc(int,max_nr_class);
    auto *count = Malloc(double,max_nr_class);

    int i;
    for(i=0;i<l;i++)
    {
      auto this_label = (int)prob->y[i];
      int j;
      for(j=0;j<nr_class;j++)
        if(this_label == label[j])
        {
          count[j] += prob->W[i];
          break;
        }
      if(j == nr_class)
      {
        if(nr_class == max_nr_class)
        {
          max_nr_class *= 2;
          label = (int *)realloc(label,max_nr_class*sizeof(int));
          count = (double *)realloc(count,max_nr_class*sizeof(double));
        }
        label[nr_class] = this_label;
        count[nr_class] = prob->W[i];
        ++nr_class;
      }
    }

    for(i=0;i<nr_class;i++)
    {
      double n1 = count[i];
      for(int j=i+1;j<nr_class;j++)
      {
        double n2 = count[j];
        if(param->nu*(n1+n2)/2 > min(n1,n2))
        {
          free(label);
          free(count);
          return "specified nu is infeasible";
        }
      }
    }
    free(label);
    free(count);
  }

  return nullptr;
}

int svm_check_probability_model(const svm_model *model)
{
  return ((model->param.svm_type == C_SVC || model->param.svm_type == NU_SVC) &&
          model->probA!=nullptr && model->probB!=nullptr) ||
      ((model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) &&
       model->probA!=nullptr);
}

void svm_set_print_string_function(void (*print_func)(const char *))
{
  if(print_func == nullptr)
    svm_print_string = &print_string_stdout;
  else
    svm_print_string = print_func;
}