```// Copyright (C) 2008  Davis E. King (davis@dlib.net)
#ifndef DLIB_KRLs_
#define DLIB_KRLs_

#include <vector>

#include "krls_abstract.h"
#include "../matrix.h"
#include "function.h"
#include "../std_allocator.h"

namespace dlib
{

// ----------------------------------------------------------------------------------------

template <typename kernel_type>
class krls
{
/*!
This is an implementation of the kernel recursive least squares algorithm described in the paper:
The Kernel Recursive Least Squares Algorithm by Yaakov Engel.
!*/

public:
typedef typename kernel_type::scalar_type scalar_type;
typedef typename kernel_type::sample_type sample_type;
typedef typename kernel_type::mem_manager_type mem_manager_type;

explicit krls (
const kernel_type& kernel_,
scalar_type tolerance_ = 0.001,
unsigned long max_dictionary_size_ = 1000000
) :
kernel(kernel_),
my_tolerance(tolerance_),
my_max_dictionary_size(max_dictionary_size_)
{
// make sure requires clause is not broken
DLIB_ASSERT(tolerance_ >= 0,
"\tkrls::krls()"
<< "\n\t You have to give a positive tolerance"
<< "\n\t this: " << this
<< "\n\t tolerance: " << tolerance_
);

clear_dictionary();
}

scalar_type tolerance() const
{
return my_tolerance;
}

unsigned long max_dictionary_size() const
{
return my_max_dictionary_size;
}

const kernel_type& get_kernel (
) const
{
return kernel;
}

void clear_dictionary ()
{
dictionary.clear();
alpha.clear();

K_inv.set_size(0,0);
K.set_size(0,0);
P.set_size(0,0);
}

scalar_type operator() (
const sample_type& x
) const
{
scalar_type temp = 0;
for (unsigned long i = 0; i < alpha.size(); ++i)
temp += alpha[i]*kern(dictionary[i], x);

return temp;
}

void train (
const sample_type& x,
scalar_type y
)
{
const scalar_type kx = kern(x,x);
if (alpha.size() == 0)
{
// just ignore this sample if it is the zero vector (or really close to being zero)
if (std::abs(kx) > std::numeric_limits<scalar_type>::epsilon())
{
// set initial state since this is the first training example we have seen

K_inv.set_size(1,1);
K_inv(0,0) = 1/kx;
K.set_size(1,1);
K(0,0) = kx;

alpha.push_back(y/kx);
dictionary.push_back(x);
P.set_size(1,1);
P(0,0) = 1;
}
}
else
{
// fill in k
k.set_size(alpha.size());
for (long r = 0; r < k.nr(); ++r)
k(r) = kern(x,dictionary[r]);

// compute the error we would have if we approximated the new x sample
// with the dictionary.  That is, do the ALD test from the KRLS paper.
a = K_inv*k;
scalar_type delta = kx - trans(k)*a;

// if this new vector isn't approximately linearly dependent on the vectors
// in our dictionary.
if (delta > my_tolerance)
{
if (dictionary.size() >= my_max_dictionary_size)
{
// We need to remove one of the old members of the dictionary before
// we proceed with adding a new one.  So remove the oldest one.
remove_dictionary_vector(0);

// recompute these guys since they were computed with the old
// kernel matrix
k = remove_row(k,0);
a = K_inv*k;
delta = kx - trans(k)*a;
}

// add x to the dictionary
dictionary.push_back(x);

// update K_inv by computing the new one in the temp matrix (equation 3.14)
matrix<scalar_type,0,0,mem_manager_type> temp(K_inv.nr()+1, K_inv.nc()+1);
// update the middle part of the matrix
set_subm(temp, get_rect(K_inv)) = K_inv + a*trans(a)/delta;
// update the right column of the matrix
set_subm(temp, 0, K_inv.nr(),K_inv.nr(),1) = -a/delta;
// update the bottom row of the matrix
set_subm(temp, K_inv.nr(), 0, 1, K_inv.nr()) = trans(-a/delta);
// update the bottom right corner of the matrix
temp(K_inv.nr(), K_inv.nc()) = 1/delta;
// put temp into K_inv
temp.swap(K_inv);

// update K (the kernel matrix)
temp.set_size(K.nr()+1, K.nc()+1);
set_subm(temp, get_rect(K)) = K;
// update the right column of the matrix
set_subm(temp, 0, K.nr(),K.nr(),1) = k;
// update the bottom row of the matrix
set_subm(temp, K.nr(), 0, 1, K.nr()) = trans(k);
temp(K.nr(), K.nc()) = kx;
// put temp into K
temp.swap(K);

// Now update the P matrix (equation 3.15)
temp.set_size(P.nr()+1, P.nc()+1);
set_subm(temp, get_rect(P)) = P;
// initialize the new sides of P
set_rowm(temp,P.nr()) = 0;
set_colm(temp,P.nr()) = 0;
temp(P.nr(), P.nc()) = 1;
temp.swap(P);

// now update the alpha vector (equation 3.16)
const scalar_type k_a = (y-trans(k)*mat(alpha))/delta;
for (unsigned long i = 0; i < alpha.size(); ++i)
{
alpha[i] -= a(i)*k_a;
}
alpha.push_back(k_a);
}
else
{
q = P*a/(1+trans(a)*P*a);

// update P (equation 3.12)
temp_matrix = trans(a)*P;
P -= q*temp_matrix;

// update the alpha vector (equation 3.13)
const scalar_type k_a = y-trans(k)*mat(alpha);
for (unsigned long i = 0; i < alpha.size(); ++i)
{
alpha[i] += (K_inv*q*k_a)(i);
}
}
}
}

void swap (
krls& item
)
{
exchange(kernel, item.kernel);
dictionary.swap(item.dictionary);
alpha.swap(item.alpha);
K_inv.swap(item.K_inv);
K.swap(item.K);
P.swap(item.P);
exchange(my_tolerance, item.my_tolerance);
q.swap(item.q);
a.swap(item.a);
k.swap(item.k);
temp_matrix.swap(item.temp_matrix);
exchange(my_max_dictionary_size, item.my_max_dictionary_size);
}

unsigned long dictionary_size (
) const { return dictionary.size(); }

decision_function<kernel_type> get_decision_function (
) const
{
return decision_function<kernel_type>(
mat(alpha),
-sum(mat(alpha))*tau,
kernel,
mat(dictionary)
);
}

friend void serialize(const krls& item, std::ostream& out)
{
serialize(item.kernel, out);
serialize(item.dictionary, out);
serialize(item.alpha, out);
serialize(item.K_inv, out);
serialize(item.K, out);
serialize(item.P, out);
serialize(item.my_tolerance, out);
serialize(item.my_max_dictionary_size, out);
}

friend void deserialize(krls& item, std::istream& in)
{
deserialize(item.kernel, in);
deserialize(item.dictionary, in);
deserialize(item.alpha, in);
deserialize(item.K_inv, in);
deserialize(item.K, in);
deserialize(item.P, in);
deserialize(item.my_tolerance, in);
deserialize(item.my_max_dictionary_size, in);
}

private:

inline scalar_type kern (const sample_type& m1, const sample_type& m2) const
{
return kernel(m1,m2) + tau;
}

void remove_dictionary_vector (
long i
)
/*!
requires
- 0 <= i < dictionary.size()
ensures
- #dictionary.size() == dictionary.size() - 1
- #alpha.size() == alpha.size() - 1
- updates the K_inv matrix so that it is still a proper inverse of the
kernel matrix
- also removes the necessary row and column from the K matrix
- uses the this->a variable so after this function runs that variable
will contain a different value.
!*/
{
// remove the dictionary vector
dictionary.erase(dictionary.begin()+i);

// remove the i'th vector from the inverse kernel matrix.  This formula is basically
// just the reverse of the way K_inv is updated by equation 3.14 during normal training.
K_inv = removerc(K_inv,i,i) - remove_row(colm(K_inv,i)/K_inv(i,i),i)*remove_col(rowm(K_inv,i),i);

// now compute the updated alpha values to take account that we just removed one of
// our dictionary vectors
a = (K_inv*remove_row(K,i)*mat(alpha));

// now copy over the new alpha values
alpha.resize(alpha.size()-1);
for (unsigned long k = 0; k < alpha.size(); ++k)
{
alpha[k] = a(k);
}

// update the P matrix as well
P = removerc(P,i,i);

// update the K matrix as well
K = removerc(K,i,i);
}

kernel_type kernel;

typedef std_allocator<sample_type, mem_manager_type> alloc_sample_type;
typedef std_allocator<scalar_type, mem_manager_type> alloc_scalar_type;
typedef std::vector<sample_type,alloc_sample_type> dictionary_vector_type;
typedef std::vector<scalar_type,alloc_scalar_type> alpha_vector_type;

dictionary_vector_type dictionary;
alpha_vector_type alpha;

matrix<scalar_type,0,0,mem_manager_type> K_inv;
matrix<scalar_type,0,0,mem_manager_type> K;
matrix<scalar_type,0,0,mem_manager_type> P;

scalar_type my_tolerance;
unsigned long my_max_dictionary_size;

// temp variables here just so we don't have to reconstruct them over and over.  Thus,
// they aren't really part of the state of this object.
matrix<scalar_type,0,1,mem_manager_type> q;
matrix<scalar_type,0,1,mem_manager_type> a;
matrix<scalar_type,0,1,mem_manager_type> k;
matrix<scalar_type,1,0,mem_manager_type> temp_matrix;

const static scalar_type tau;

};

template <typename kernel_type>
const typename kernel_type::scalar_type krls<kernel_type>::tau = static_cast<typename kernel_type::scalar_type>(0.01);

// ----------------------------------------------------------------------------------------

template <typename kernel_type>
void swap(krls<kernel_type>& a, krls<kernel_type>& b)
{ a.swap(b); }

// ----------------------------------------------------------------------------------------

}

#endif // DLIB_KRLs_

```