denoising auto-encoders (SdA)

去噪自编码器背后的思想很简单. 为了迫使隐藏层单元发现更多鲁棒性好的特征, 以及阻止它学习恒等函数, 我们拿受损的输入来训练自编码器重构输入.
降噪自动编码器是自动编码器的随机版. 直观地说, 一个降噪自动编码器做两件事情: 试图对输入编码( 保存输入的信息 ), 和试图消除随机应用于自动编码器输入的损坏处理的影响. 后者则只能通过输入之间的统计相关性实现. 降噪自动编码器可以从不同的角度理解。

将自编码器类转换成去噪自编码器类, 我们需要做的是给输入增加一个随机损坏(stochastic corruption)操作. 

//dA.h
class dA {

public:
  int N;
  int n_visible;
  int n_hidden;
  double **W;
  double *hbias;
  double *vbias;
  dA(int, int, int , double**, double*, double*);
  ~dA();
  void get_corrupted_input(int*, int*, double);
  void get_hidden_values(int*, double*);
  void get_reconstructed_input(double*, double*);
  void train(int*, double, double);
  void reconstruct(int*, double*);
};

//dA.cpp
#include <iostream>
#include <math.h>
#include "dA.h"
using namespace std;


double uniform(double min, double max) {
  return rand() / (RAND_MAX + 1.0) * (max - min) + min;
}

int binomial(int n, double p) {
  if(p < 0 || p > 1) return 0;
  
  int c = 0;
  double r;
  
  for(int i=0; i<n; i++) {
    r = rand() / (RAND_MAX + 1.0);
    if (r < p) c++;
  }

  return c;
}

double sigmoid(double x) {
  return 1.0 / (1.0 + exp(-x));
}


dA::dA(int size, int n_v, int n_h, double **w, double *hb, double *vb) {
  N = size;
  n_visible = n_v;
  n_hidden = n_h;

  if(w == NULL) {
    W = new double*[n_hidden];
    for(int i=0; i<n_hidden; i++) W[i] = new double[n_visible];
    double a = 1.0 / n_visible;

    for(int i=0; i<n_hidden; i++) {
      for(int j=0; j<n_visible; j++) {
        W[i][j] = uniform(-a, a);
      }
    }
  } else {
    W = w;
  }

  if(hb == NULL) {
    hbias = new double[n_hidden];
    for(int i=0; i<n_hidden; i++) hbias[i] = 0;
  } else {
    hbias = hb;
  }

  if(vb == NULL) {
    vbias = new double[n_visible];
    for(int i=0; i<n_visible; i++) vbias[i] = 0;
  } else {
    vbias = vb;
  }
}

dA::~dA() {
  for(int i=0; i<n_hidden; i++) delete[] W[i];
  delete[] W;
  delete[] hbias;
  delete[] vbias;
}

void dA::get_corrupted_input(int *x, int *tilde_x, double p) {
  for(int i=0; i<n_visible; i++) {
    if(x[i] == 0) {
      tilde_x[i] = 0;
    } else {
      tilde_x[i] = binomial(1, p);
    }
  }
}

// Encode
void dA::get_hidden_values(int *x, double *y) {
  for(int i=0; i<n_hidden; i++) {
    y[i] = 0;
    for(int j=0; j<n_visible; j++) {
      y[i] += W[i][j] * x[j];
    }
    y[i] += hbias[i];
    y[i] = sigmoid(y[i]);
  }
}

// Decode
void dA::get_reconstructed_input(double *y, double *z) {
  for(int i=0; i<n_visible; i++) {
    z[i] = 0;
    for(int j=0; j<n_hidden; j++) {
      z[i] += W[j][i] * y[j];
    }
    z[i] += vbias[i];
    z[i] = sigmoid(z[i]);
  }
}

void dA::train(int *x, double lr, double corruption_level) {
  int *tilde_x = new int[n_visible];
  double *y = new double[n_hidden];
  double *z = new double[n_visible];

  double *L_vbias = new double[n_visible];
  double *L_hbias = new double[n_hidden];

  double p = 1 - corruption_level;

  get_corrupted_input(x, tilde_x, p);
  get_hidden_values(tilde_x, y);
  get_reconstructed_input(y, z);
  
  // vbias
  for(int i=0; i<n_visible; i++) {
    L_vbias[i] = x[i] - z[i];
    vbias[i] += lr * L_vbias[i] / N;
  }

  // hbias
  for(int i=0; i<n_hidden; i++) {
    L_hbias[i] = 0;
    for(int j=0; j<n_visible; j++) {
      L_hbias[i] += W[i][j] * L_vbias[j];
    }
    L_hbias[i] *= y[i] * (1 - y[i]);

    hbias[i] += lr * L_hbias[i] / N;
  }
  
  // W
  for(int i=0; i<n_hidden; i++) {
    for(int j=0; j<n_visible; j++) {
      W[i][j] += lr * (L_hbias[i] * tilde_x[j] + L_vbias[j] * y[i]) / N;
    }
  }

  delete[] L_hbias;
  delete[] L_vbias;
  delete[] z;
  delete[] y;
  delete[] tilde_x;
}

void dA::reconstruct(int *x, double *z) {
  double *y = new double[n_hidden];

  get_hidden_values(x, y);
  get_reconstructed_input(y, z);

  delete[] y;
}



void test_dA() {
  srand(0);
  
  double learning_rate = 0.1;
  double corruption_level = 0.3;
  int training_epochs = 100;

  int train_N = 10;
  int test_N = 2;
  int n_visible = 20;
  int n_hidden = 5;

  // training data
  int train_X[10][20] = {
    {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0}
  };

  // construct dA
  dA da(train_N, n_visible, n_hidden, NULL, NULL, NULL);

  // train
  for(int epoch=0; epoch<training_epochs; epoch++) {
    for(int i=0; i<train_N; i++) {
      da.train(train_X[i], learning_rate, corruption_level);
    }
  }

  // test data
  int test_X[2][20] = {
    {1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0}
  };
  double reconstructed_X[2][20];


  // test
  for(int i=0; i<test_N; i++) {
    da.reconstruct(test_X[i], reconstructed_X[i]);
    for(int j=0; j<n_visible; j++) {
      printf("%.5f ", reconstructed_X[i][j]);
    }
    cout << endl;
  }

  cout << endl;
}



int main() {
  test_dA();
  return 0;
}

//SdA.h
class SdA {

public:
  int N;
  int n_ins;
  int *hidden_layer_sizes;
  int n_outs;
  int n_layers;
  HiddenLayer **sigmoid_layers;
  dA **dA_layers;
  LogisticRegression *log_layer;
  SdA(int, int, int*, int, int);
  ~SdA();
  void pretrain(int*, double, double, int);
  void finetune(int*, int*, double, int);
  void predict(int*, double*);
};

//SdA.cpp
#include <iostream>
#include <math.h>
#include "HiddenLayer.h"
#include "dA.h"
#include "LogisticRegression.h"
#include "SdA.h"
using namespace std;

double uniform(double min, double max) {
  return rand() / (RAND_MAX + 1.0) * (max - min) + min;
}

int binomial(int n, double p) {
  if(p < 0 || p > 1) return 0;
  
  int c = 0;
  double r;
  
  for(int i=0; i<n; i++) {
    r = rand() / (RAND_MAX + 1.0);
    if (r < p) c++;
  }

  return c;
}

double sigmoid(double x) {
  return 1.0 / (1.0 + exp(-x));
}


// SdA
SdA::SdA(int size, int n_i, int *hls, int n_o, int n_l) {
  int input_size;

  N = size;
  n_ins = n_i;
  hidden_layer_sizes = hls;
  n_outs = n_o;
  n_layers = n_l;

  sigmoid_layers = new HiddenLayer*[n_layers];
  dA_layers = new dA*[n_layers];

  // construct multi-layer
  for(int i=0; i<n_layers; i++) {
    if(i == 0) {
      input_size = n_ins;
    } else {
      input_size = hidden_layer_sizes[i-1];
    }

    // construct sigmoid_layer
    sigmoid_layers[i] = new HiddenLayer(N, input_size, hidden_layer_sizes[i], NULL, NULL);

    // construct dA_layer
    dA_layers[i] = new dA(N, input_size, hidden_layer_sizes[i],\
                          sigmoid_layers[i]->W, sigmoid_layers[i]->b, NULL);
  }

  // layer for output using LogisticRegression
  log_layer = new LogisticRegression(N, hidden_layer_sizes[n_layers-1], n_outs);
}

SdA::~SdA() {
  delete log_layer;

  for(int i=0; i<n_layers; i++) {
    delete sigmoid_layers[i];
    delete dA_layers[i];
  }
  delete[] sigmoid_layers;
  delete[] dA_layers;
}

void SdA::pretrain(int *input, double lr, double corruption_level, int epochs) {
  int *layer_input;
  int prev_layer_input_size;
  int *prev_layer_input;

  int *train_X = new int[n_ins];

  for(int i=0; i<n_layers; i++) {  // layer-wise

    for(int epoch=0; epoch<epochs; epoch++) {  // training epochs

      for(int n=0; n<N; n++) { // input x1...xN
        // initial input
        for(int m=0; m<n_ins; m++) train_X[m] = input[n * n_ins + m];

        // layer input
        for(int l=0; l<=i; l++) {

          if(l == 0) {
            layer_input = new int[n_ins];
            for(int j=0; j<n_ins; j++) layer_input[j] = train_X[j];
          } else {
            if(l == 1) prev_layer_input_size = n_ins;
            else prev_layer_input_size = hidden_layer_sizes[l-2];

            prev_layer_input = new int[prev_layer_input_size];
            for(int j=0; j<prev_layer_input_size; j++) prev_layer_input[j] = layer_input[j];
            delete[] layer_input;

            layer_input = new int[hidden_layer_sizes[l-1]];

            sigmoid_layers[l-1]->sample_h_given_v(prev_layer_input, layer_input);
            delete[] prev_layer_input;
          }
        }

        dA_layers[i]->train(layer_input, lr, corruption_level);

      }
    }
  }

  delete[] train_X;
  delete[] layer_input;
}

void SdA::finetune(int *input, int *label, double lr, int epochs) {
  int *layer_input;
  int prev_layer_input_size;
  int *prev_layer_input;

  int *train_X = new int[n_ins];
  int *train_Y = new int[n_outs];

  for(int epoch=0; epoch<epochs; epoch++) {
    for(int n=0; n<N; n++) { // input x1...xN
      // initial input
      for(int m=0; m<n_ins; m++)  train_X[m] = input[n * n_ins + m];
      for(int m=0; m<n_outs; m++) train_Y[m] = label[n * n_outs + m];

      // layer input
      for(int i=0; i<n_layers; i++) {
        if(i == 0) {
          prev_layer_input = new int[n_ins];
          for(int j=0; j<n_ins; j++) prev_layer_input[j] = train_X[j];
        } else {
          prev_layer_input = new int[hidden_layer_sizes[i-1]];
          for(int j=0; j<hidden_layer_sizes[i-1]; j++) prev_layer_input[j] = layer_input[j];
          delete[] layer_input;
        }


        layer_input = new int[hidden_layer_sizes[i]];
        sigmoid_layers[i]->sample_h_given_v(prev_layer_input, layer_input);
        delete[] prev_layer_input;
      }

      log_layer->train(layer_input, train_Y, lr);
    }
    // lr *= 0.95;
  }

  delete[] layer_input;
  delete[] train_X;
  delete[] train_Y;
}

void SdA::predict(int *x, double *y) {
  double *layer_input;
  int prev_layer_input_size;
  double *prev_layer_input;

  double linear_output;

  prev_layer_input = new double[n_ins];
  for(int j=0; j<n_ins; j++) prev_layer_input[j] = x[j];

  // layer activation
  for(int i=0; i<n_layers; i++) {
    layer_input = new double[sigmoid_layers[i]->n_out];

    for(int k=0; k<sigmoid_layers[i]->n_out; k++) {
      linear_output = 0.0;

      for(int j=0; j<sigmoid_layers[i]->n_in; j++) {
        linear_output += sigmoid_layers[i]->W[k][j] * prev_layer_input[j];
      }
      linear_output += sigmoid_layers[i]->b[k];
      layer_input[k] = sigmoid(linear_output);
    }
    delete[] prev_layer_input;

    if(i < n_layers-1) {
      prev_layer_input = new double[sigmoid_layers[i]->n_out];
      for(int j=0; j<sigmoid_layers[i]->n_out; j++) prev_layer_input[j] = layer_input[j];
      delete[] layer_input;
    }
  }
  
  for(int i=0; i<log_layer->n_out; i++) {
    y[i] = 0;
    for(int j=0; j<log_layer->n_in; j++) {
      y[i] += log_layer->W[i][j] * layer_input[j];
    }
    y[i] += log_layer->b[i];
  }
  
  log_layer->softmax(y);


  delete[] layer_input;
}


// HiddenLayer
HiddenLayer::HiddenLayer(int size, int in, int out, double **w, double *bp) {
  N = size;
  n_in = in;
  n_out = out;

  if(w == NULL) {
    W = new double*[n_out];
    for(int i=0; i<n_out; i++) W[i] = new double[n_in];
    double a = 1.0 / n_in;

    for(int i=0; i<n_out; i++) {
      for(int j=0; j<n_in; j++) {
        W[i][j] = uniform(-a, a);
      }
    }
  } else {
    W = w;
  }

  if(bp == NULL) {
    b = new double[n_out];
  } else {
    b = bp;
  }
}

HiddenLayer::~HiddenLayer() {
  for(int i=0; i<n_out; i++) delete W[i];
  delete[] W;
  delete[] b;
}

double HiddenLayer::output(int *input, double *w, double b) {
  double linear_output = 0.0;
  for(int j=0; j<n_in; j++) {
    linear_output += w[j] * input[j];
  }
  linear_output += b;
  return sigmoid(linear_output);
}

void HiddenLayer::sample_h_given_v(int *input, int *sample) {
  for(int i=0; i<n_out; i++) {
    sample[i] = binomial(1, output(input, W[i], b[i]));
  }
}


// dA
dA::dA(int size, int n_v, int n_h, double **w, double *hb, double *vb) {
  N = size;
  n_visible = n_v;
  n_hidden = n_h;

  if(w == NULL) {
    W = new double*[n_hidden];
    for(int i=0; i<n_hidden; i++) W[i] = new double[n_visible];
    double a = 1.0 / n_visible;

    for(int i=0; i<n_hidden; i++) {
      for(int j=0; j<n_visible; j++) {
        W[i][j] = uniform(-a, a);
      }
    }
  } else {
    W = w;
  }

  if(hb == NULL) {
    hbias = new double[n_hidden];
    for(int i=0; i<n_hidden; i++) hbias[i] = 0;
  } else {
    hbias = hb;
  }

  if(vb == NULL) {
    vbias = new double[n_visible];
    for(int i=0; i<n_visible; i++) vbias[i] = 0;
  } else {
    vbias = vb;
  }
}

dA::~dA() {
  // for(int i=0; i<n_hidden; i++) delete[] W[i];
  // delete[] W;
  // delete[] hbias;
  delete[] vbias;
}

void dA::get_corrupted_input(int *x, int *tilde_x, double p) {
  for(int i=0; i<n_visible; i++) {
    if(x[i] == 0) {
      tilde_x[i] = 0;
    } else {
      tilde_x[i] = binomial(1, p);
    }
  }
}

// Encode
void dA::get_hidden_values(int *x, double *y) {
  for(int i=0; i<n_hidden; i++) {
    y[i] = 0;
    for(int j=0; j<n_visible; j++) {
      y[i] += W[i][j] * x[j];
    }
    y[i] += hbias[i];
    y[i] = sigmoid(y[i]);
  }
}

// Decode
void dA::get_reconstructed_input(double *y, double *z) {
  for(int i=0; i<n_visible; i++) {
    z[i] = 0;
    for(int j=0; j<n_hidden; j++) {
      z[i] += W[j][i] * y[j];
    }
    z[i] += vbias[i];
    z[i] = sigmoid(z[i]);
  }
}

void dA::train(int *x, double lr, double corruption_level) {
  int *tilde_x = new int[n_visible];
  double *y = new double[n_hidden];
  double *z = new double[n_visible];

  double *L_vbias = new double[n_visible];
  double *L_hbias = new double[n_hidden];

  double p = 1 - corruption_level;

  get_corrupted_input(x, tilde_x, p);
  get_hidden_values(tilde_x, y);
  get_reconstructed_input(y, z);
  
  // vbias
  for(int i=0; i<n_visible; i++) {
    L_vbias[i] = x[i] - z[i];
    vbias[i] += lr * L_vbias[i] / N;
  }

  // hbias
  for(int i=0; i<n_hidden; i++) {
    L_hbias[i] = 0;
    for(int j=0; j<n_visible; j++) {
      L_hbias[i] += W[i][j] * L_vbias[j];
    }
    L_hbias[i] *= y[i] * (1 - y[i]);

    hbias[i] += lr * L_hbias[i] / N;
  }
  
  // W
  for(int i=0; i<n_hidden; i++) {
    for(int j=0; j<n_visible; j++) {
      W[i][j] += lr * (L_hbias[i] * tilde_x[j] + L_vbias[j] * y[i]) / N;
    }
  }

  delete[] L_hbias;
  delete[] L_vbias;
  delete[] z;
  delete[] y;
  delete[] tilde_x;
}

void dA::reconstruct(int *x, double *z) {
  double *y = new double[n_hidden];

  get_hidden_values(x, y);
  get_reconstructed_input(y, z);

  delete[] y;
}


// LogisticRegression
LogisticRegression::LogisticRegression(int size, int in, int out) {
  N = size;
  n_in = in;
  n_out = out;

  W = new double*[n_out];
  for(int i=0; i<n_out; i++) W[i] = new double[n_in];
  b = new double[n_out];

  for(int i=0; i<n_out; i++) {
    for(int j=0; j<n_in; j++) {
      W[i][j] = 0;
    }
    b[i] = 0;
  }
}

LogisticRegression::~LogisticRegression() {
  for(int i=0; i<n_out; i++) delete[] W[i];
  delete[] W;
  delete[] b;
}


void LogisticRegression::train(int *x, int *y, double lr) {
  double *p_y_given_x = new double[n_out];
  double *dy = new double[n_out];

  for(int i=0; i<n_out; i++) {
    p_y_given_x[i] = 0;
    for(int j=0; j<n_in; j++) {
      p_y_given_x[i] += W[i][j] * x[j];
    }
    p_y_given_x[i] += b[i];
  }
  softmax(p_y_given_x);

  for(int i=0; i<n_out; i++) {
    dy[i] = y[i] - p_y_given_x[i];

    for(int j=0; j<n_in; j++) {
      W[i][j] += lr * dy[i] * x[j] / N;
    }

    b[i] += lr * dy[i] / N;
  }
  
  delete[] p_y_given_x;
  delete[] dy;
}

void LogisticRegression::softmax(double *x) {
  double max = 0.0;
  double sum = 0.0;
  
  for(int i=0; i<n_out; i++) if(max < x[i]) max = x[i];
  for(int i=0; i<n_out; i++) {
    x[i] = exp(x[i] - max);
    sum += x[i];
  } 

  for(int i=0; i<n_out; i++) x[i] /= sum;
}

void LogisticRegression::predict(int *x, double *y) {
  for(int i=0; i<n_out; i++) {
    y[i] = 0;
    for(int j=0; j<n_in; j++) {
      y[i] += W[i][j] * x[j];
    }
    y[i] += b[i];
  }

  softmax(y);
}


void test_sda() {
  srand(0);

  double pretrain_lr = 0.1;
  double corruption_level = 0.3;
  int pretraining_epochs = 1000;
  double finetune_lr = 0.1;
  int finetune_epochs = 500;

  int train_N = 10;
  int test_N = 4;
  int n_ins = 28;
  int n_outs = 2;
  int hidden_layer_sizes[] = {15, 15};
  int n_layers = sizeof(hidden_layer_sizes) / sizeof(hidden_layer_sizes[0]);

  // training data
  int train_X[10][28] = {
    {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1}
  };

  int train_Y[10][2] = {
    {1, 0},
    {1, 0},
    {1, 0},
    {1, 0},
    {1, 0},
    {0, 1},
    {0, 1},
    {0, 1},
    {0, 1},
    {0, 1}
  };

  // construct SdA
  SdA sda(train_N, n_ins, hidden_layer_sizes, n_outs, n_layers);

  // pretrain
  sda.pretrain(*train_X, pretrain_lr, corruption_level, pretraining_epochs);

  // finetune
  sda.finetune(*train_X, *train_Y, finetune_lr, finetune_epochs);


  // test data
  int test_X[4][28] = {
    {1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1}
  };

  double test_Y[4][28];

  // test
  for(int i=0; i<test_N; i++) {
    sda.predict(test_X[i], test_Y[i]);
    for(int j=0; j<n_outs; j++) {
      printf("%.5f ", test_Y[i][j]);
    }
    cout << endl;
  }
  
}


int main() {
  test_sda();
  return 0;
}