PLearn 0.1
Public Member Functions | Static Public Member Functions | Static Public Attributes | Protected Member Functions | Static Protected Member Functions | Private Types | Private Member Functions
PLearn::SquaredExponentialARDKernel Class Reference

Squared-Exponential kernel that can be used for Automatic Relevance Determination. More...

#include <SquaredExponentialARDKernel.h>

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List of all members.

Public Member Functions

 SquaredExponentialARDKernel ()
 Default constructor.
virtual real evaluate (const Vec &x1, const Vec &x2) const
 Compute K(x1,x2).
virtual void computeGramMatrix (Mat K) const
 Compute the Gram Matrix.
virtual void computeGramMatrixDerivative (Mat &KD, const string &kernel_param, real epsilon=1e-6) const
 Directly compute the derivative with respect to hyperparameters (Faster than finite differences...)
virtual void evaluate_all_i_x (const Vec &x, const Vec &k_xi_x, real squared_norm_of_x=-1, int istart=0) const
 Fill k_xi_x with K(x_i, x), for all i from istart to istart + k_xi_x.length() - 1.
virtual string classname () const
virtual OptionListgetOptionList () const
virtual OptionMapgetOptionMap () const
virtual RemoteMethodMapgetRemoteMethodMap () const
virtual
SquaredExponentialARDKernel
deepCopy (CopiesMap &copies) const
virtual void build ()
 Post-constructor.
virtual void makeDeepCopyFromShallowCopy (CopiesMap &copies)
 Transforms a shallow copy into a deep copy.

Static Public Member Functions

static string _classname_ ()
static OptionList_getOptionList_ ()
static RemoteMethodMap_getRemoteMethodMap_ ()
static Object_new_instance_for_typemap_ ()
static bool _isa_ (const Object *o)
static void _static_initialize_ ()
static const PPathdeclaringFile ()

Static Public Attributes

static StaticInitializer _static_initializer_

Protected Member Functions

real derivIspSignalSigma (int i, int j, int arg, real K) const
 Derivative function with respect to isp_signal_sigma.
real derivIspGlobalSigma (int i, int j, int arg, real K) const
 Derivative function with respect to isp_global_sigma.
void computeGramMatrixDerivIspSignalSigma (Mat &KD) const
void computeGramMatrixDerivIspInputSigma (Mat &KD, int arg) const

Static Protected Member Functions

static void declareOptions (OptionList &ol)
 Declares the class options.

Private Types

typedef ARDBaseKernel inherited

Private Member Functions

void build_ ()
 This does the actual building.

Detailed Description

Squared-Exponential kernel that can be used for Automatic Relevance Determination.

This is a variant of the GaussianKernel (a.k.a. Radial Basis Function) that provides a different length-scale parameter for each input variable. When used in conjunction with GaussianProcessRegressor, this kernel may be used for Automatic Relevance Determination (ARD), a procedure wherein the significance of each input variable for the prediction task is found automatically through numerical optimization.

Similar to C.E. Rasmussen's GPML code (see http://www.gaussianprocess.org), this kernel function is specified as:

k(x,y) = sf * exp(- 0.5 * (sum_i (x_i - y_i)^2 / w_i)) * k_kron(x,y)

where sf is softplus(isp_signal_sigma), w_i is softplus(isp_global_sigma + isp_input_sigma[i]), and k_kron(x,y) is the result of the KroneckerBaseKernel evaluation, or 1.0 if there are no Kronecker terms. Note that since the Kronecker terms are incorporated multiplicatively, the very presence of the term associated to this kernel can be gated by the value of some input variable(s) (that are incorporated within one or more Kronecker terms).

Note that contrarily to previous versions that incorporated IID noise and Kronecker terms ADDITIVELY, this version does not add any noise at all (and as explained above incorporates the Kronecker terms multiplicatively). For best results, especially with moderately noisy data, IT IS IMPERATIVE to use whis kernel within a SummationKernel in conjunction with an IIDNoiseKernel, as follows (e.g. within a GaussianProcessRegressor):

kernel = SummationKernel(terms = [ SquaredExponentialARDKernel(), IIDNoiseKernel() ] )

Note that to make its operations more robust when used with unconstrained optimization of hyperparameters, all hyperparameters of this kernel are specified in the inverse softplus domain. See IIDNoiseKernel for more explanations.

Definition at line 86 of file SquaredExponentialARDKernel.h.


Member Typedef Documentation

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 88 of file SquaredExponentialARDKernel.h.


Constructor & Destructor Documentation

PLearn::SquaredExponentialARDKernel::SquaredExponentialARDKernel ( )

Default constructor.

Definition at line 85 of file SquaredExponentialARDKernel.cc.

{ }

Member Function Documentation

string PLearn::SquaredExponentialARDKernel::_classname_ ( ) [static]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

OptionList & PLearn::SquaredExponentialARDKernel::_getOptionList_ ( ) [static]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

RemoteMethodMap & PLearn::SquaredExponentialARDKernel::_getRemoteMethodMap_ ( ) [static]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

bool PLearn::SquaredExponentialARDKernel::_isa_ ( const Object o) [static]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

Object * PLearn::SquaredExponentialARDKernel::_new_instance_for_typemap_ ( ) [static]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

StaticInitializer SquaredExponentialARDKernel::_static_initializer_ & PLearn::SquaredExponentialARDKernel::_static_initialize_ ( ) [static]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

void PLearn::SquaredExponentialARDKernel::build ( ) [virtual]

Post-constructor.

The normal implementation should call simply inherited::build(), then this class's build_(). This method should be callable again at later times, after modifying some option fields to change the "architecture" of the object.

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 100 of file SquaredExponentialARDKernel.cc.

{
    // ### Nothing to add here, simply calls build_
    inherited::build();
    build_();
}
void PLearn::SquaredExponentialARDKernel::build_ ( ) [private]

This does the actual building.

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 110 of file SquaredExponentialARDKernel.cc.

{
    // Ensure that we multiply in Kronecker terms
    inherited::m_default_value = 1.0;
}
string PLearn::SquaredExponentialARDKernel::classname ( ) const [virtual]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

void PLearn::SquaredExponentialARDKernel::computeGramMatrix ( Mat  K) const [virtual]

Compute the Gram Matrix.

Reimplemented from PLearn::KroneckerBaseKernel.

Definition at line 160 of file SquaredExponentialARDKernel.cc.

References PLearn::diff(), PLearn::exp(), i, j, m, PLearn::TMat< T >::mod(), n, PLASSERT, PLearn::TMat< T >::size(), and PLearn::softplus().

{
    PLASSERT( !m_isp_input_sigma.size() || dataInputsize() == m_isp_input_sigma.size() );
    PLASSERT( K.size() == 0 || m_data_cache.size() > 0 );  // Ensure data cached OK

    // Compute Kronecker gram matrix
    inherited::computeGramMatrix(K);

    // Precompute some terms. Make sure that the input sigmas don't get too
    // small
    real sf    = softplus(m_isp_signal_sigma);
    m_input_sigma.resize(dataInputsize());
    softplusFloor(m_isp_global_sigma, 1e-6);
    m_input_sigma.fill(m_isp_global_sigma);  // Still in ISP domain
    for (int i=0, n=m_input_sigma.size() ; i<n ; ++i) {
        if (m_isp_input_sigma.size() > 0) {
            softplusFloor(m_isp_input_sigma[i], 1e-6);
            m_input_sigma[i] += m_isp_input_sigma[i];
        }
        m_input_sigma[i] = softplus(m_input_sigma[i]);
    }

    // Compute Gram Matrix
    int  l = data->length();
    int  m = K.mod();
    int  n = dataInputsize();
    int  cache_mod = m_data_cache.mod();

    real *data_start = &m_data_cache(0,0);
    real *Ki = K[0];                         // Start of current row
    real *Kij;                               // Current element along row
    real *input_sigma_data = m_input_sigma.data();
    real *xi = data_start;
    
    for (int i=0 ; i<l ; ++i, xi += cache_mod, Ki+=m)
    {
        Kij = Ki;
        real *xj = data_start;

        for (int j=0; j<=i; ++j, xj += cache_mod) {
            // Kernel evaluation per se
            real *x1 = xi;
            real *x2 = xj;
            real *p_inpsigma = input_sigma_data;
            real sum_wt = 0.0;
            int  k = n;

            // Use Duff's device to unroll the following loop:
            //     while (k--) {
            //         real diff = *x1++ - *x2++;
            //         sum_wt += (diff * diff) / *p_inpsigma++;
            //     }
            real diff;
            switch (k % 8) {
            case 0: do { diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
            case 7:      diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
            case 6:      diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
            case 5:      diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
            case 4:      diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
            case 3:      diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
            case 2:      diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
            case 1:      diff = *x1++ - *x2++; sum_wt += (diff*diff) / *p_inpsigma++;
                       } while((k -= 8) > 0);
            }

            // Multiplicatively update kernel matrix (already pre-filled with
            // Kronecker terms, or 1.0 if no Kronecker terms, as per build_).
            real Kij_cur = *Kij * sf * exp(-0.5 * sum_wt);
            *Kij++ = Kij_cur;
        }
    }
    if (cache_gram_matrix) {
        gram_matrix.resize(l,l);
        gram_matrix << K;
        gram_matrix_is_cached = true;
    }
}

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void PLearn::SquaredExponentialARDKernel::computeGramMatrixDerivative ( Mat KD,
const string &  kernel_param,
real  epsilon = 1e-6 
) const [virtual]

Directly compute the derivative with respect to hyperparameters (Faster than finite differences...)

Reimplemented from PLearn::Kernel.

Definition at line 241 of file SquaredExponentialARDKernel.cc.

References derivIspGlobalSigma(), PLASSERT, PLearn::string_begins_with(), and PLearn::tolong().

{
    static const string ISS("isp_signal_sigma");
    static const string IGS("isp_global_sigma");
    static const string IIS("isp_input_sigma[");

    if (kernel_param == ISS) {
        computeGramMatrixDerivIspSignalSigma(KD);
        
        // computeGramMatrixDerivNV<
        //     SquaredExponentialARDKernel,
        //     &SquaredExponentialARDKernel::derivIspSignalSigma>(KD, this, -1);
    }
    else if (kernel_param == IGS) {
        computeGramMatrixDerivNV<
            SquaredExponentialARDKernel,
            &SquaredExponentialARDKernel::derivIspGlobalSigma>(KD, this, -1);
    }
    else if (string_begins_with(kernel_param, IIS) &&
             kernel_param[kernel_param.size()-1] == ']')
    {
        int arg = tolong(kernel_param.substr(
                             IIS.size(), kernel_param.size() - IIS.size() - 1));
        PLASSERT( arg < m_isp_input_sigma.size() );

        computeGramMatrixDerivIspInputSigma(KD, arg);

    }
    else
        inherited::computeGramMatrixDerivative(KD, kernel_param, epsilon);
}

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void PLearn::SquaredExponentialARDKernel::computeGramMatrixDerivIspInputSigma ( Mat KD,
int  arg 
) const [protected]

Definition at line 328 of file SquaredExponentialARDKernel.cc.

References PLearn::TMat< T >::data(), PLearn::diff(), i, j, PLearn::TMat< T >::mod(), PLASSERT_MSG, PLearn::TMat< T >::resize(), and PLearn::sigmoid().

{
    // Precompute some terms
    real input_sigma_arg = m_input_sigma[arg];
    real input_sigma_sq  = input_sigma_arg * input_sigma_arg;
    real input_sigmoid   = sigmoid(m_isp_global_sigma + m_isp_input_sigma[arg]);
    
    // Compute Gram Matrix derivative w.r.t. isp_input_sigma[arg]
    int  l = data->length();
    PLASSERT_MSG(
        gram_matrix.width() == l && gram_matrix.length() == l,
        "To compute the derivative with respect to 'isp_input_sigma[i]', the\n"
        "Gram matrix must be precomputed and cached in SquaredExponentialARDKernel.");

    // Variables that walk over the data matrix
    int  cache_mod = m_data_cache.mod();
    real *data_start = &m_data_cache(0,0);
    real *xi = data_start+arg;               // Iterator on data rows

    // Variables that walk over the gram cache
    int   gram_cache_mod = gram_matrix.mod();
    real *gram_cache_row = gram_matrix.data();
    real *gram_cache_cur;
    
    // Variables that walk over the kernel derivative matrix (KD)
    KD.resize(l,l);
    real* KDi = KD.data();                   // Start of row i
    real* KDij;                              // Current element on row i
    int   KD_mod = KD.mod();

    // Iterate on rows of derivative matrix
    for (int i=0 ; i<l ; ++i, xi += cache_mod, KDi += KD_mod,
             gram_cache_row += gram_cache_mod)
    {
        KDij = KDi;
        real *xj  = data_start+arg;           // Inner iterator on data rows
        gram_cache_cur = gram_cache_row;

        // Iterate on columns of derivative matrix
        for (int j=0 ; j <= i
                 ; ++j, xj += cache_mod, ++gram_cache_cur)
        {
            real diff    = *xi - *xj;
            real sq_diff = diff * diff;
            real KD_cur  = 0.5 * *gram_cache_cur *
                           input_sigmoid * sq_diff / input_sigma_sq;

            // Set into derivative matrix
            *KDij++ = KD_cur;
        }
    }
}

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void PLearn::SquaredExponentialARDKernel::computeGramMatrixDerivIspSignalSigma ( Mat KD) const [protected]

Definition at line 312 of file SquaredExponentialARDKernel.cc.

References PLASSERT_MSG, PLearn::TMat< T >::resize(), PLearn::sigmoid(), and PLearn::softplus().

{
    int l = data->length();
    KD.resize(l,l);
    PLASSERT_MSG(
        gram_matrix.width() == l && gram_matrix.length() == l,
        "To compute the derivative with respect to 'isp_signal_sigma', the\n"
        "Gram matrix must be precomputed and cached in SquaredExponentialARDKernel.");
    
    KD << gram_matrix;
    KD *= sigmoid(m_isp_signal_sigma)/softplus(m_isp_signal_sigma);
}

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void PLearn::SquaredExponentialARDKernel::declareOptions ( OptionList ol) [static, protected]

Declares the class options.

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 91 of file SquaredExponentialARDKernel.cc.

{
    // Now call the parent class' declareOptions
    inherited::declareOptions(ol);
}
static const PPath& PLearn::SquaredExponentialARDKernel::declaringFile ( ) [inline, static]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 123 of file SquaredExponentialARDKernel.h.

:
SquaredExponentialARDKernel * PLearn::SquaredExponentialARDKernel::deepCopy ( CopiesMap copies) const [virtual]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

real PLearn::SquaredExponentialARDKernel::derivIspGlobalSigma ( int  i,
int  j,
int  arg,
real  K 
) const [protected]

Derivative function with respect to isp_global_sigma.

Definition at line 295 of file SquaredExponentialARDKernel.cc.

References PLearn::fast_is_equal(), pl_log, PLearn::sigmoid(), and PLearn::softplus().

Referenced by computeGramMatrixDerivative().

{
    if (fast_is_equal(K,0.))
        return 0.;

    // The norm term inside the exponential may be accessed as Log(K/sf)
    real inner = pl_log(K / softplus(m_isp_signal_sigma));
    return - K * inner * sigmoid(m_isp_global_sigma) / softplus(m_isp_global_sigma);

    // Note: in the above expression for 'inner' there is the implicit
    // assumption that the input_sigma[i] are zero, which allows the
    // sigmoid/softplus term to be factored out of the norm summation.
}

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real PLearn::SquaredExponentialARDKernel::derivIspSignalSigma ( int  i,
int  j,
int  arg,
real  K 
) const [protected]

Derivative function with respect to isp_signal_sigma.

Definition at line 286 of file SquaredExponentialARDKernel.cc.

References PLearn::sigmoid(), and PLearn::softplus().

{
    // (No longer used; see computeGramMatrixDerivIspInputSigma below)
    return K*sigmoid(m_isp_signal_sigma)/softplus(m_isp_signal_sigma);
}

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real PLearn::SquaredExponentialARDKernel::evaluate ( const Vec x1,
const Vec x2 
) const [virtual]

Compute K(x1,x2).

Reimplemented from PLearn::KroneckerBaseKernel.

Definition at line 119 of file SquaredExponentialARDKernel.cc.

References PLearn::TVec< T >::data(), PLearn::diff(), PLearn::exp(), PLearn::fast_is_equal(), i, n, PLASSERT, PLearn::TVec< T >::size(), and PLearn::softplus().

{
    PLASSERT( x1.size() == x2.size() );
    PLASSERT( !m_isp_input_sigma.size() || x1.size() == m_isp_input_sigma.size() );

    real gating_term = inherited::evaluate(x1,x2);
    if (fast_is_equal(gating_term, 0.0))
        return 0.0;
    
    if (x1.size() == 0)
        return softplus(m_isp_signal_sigma) * gating_term;
    
    const real* px1 = x1.data();
    const real* px2 = x2.data();
    real sf         = softplus(m_isp_signal_sigma);
    real expval     = 0.0;
    
    if (m_isp_input_sigma.size() > 0) {
        const real* pinpsig = m_isp_input_sigma.data();
        for (int i=0, n=x1.size() ; i<n ; ++i) {
            real diff   = *px1++ - *px2++;
            real sqdiff = diff * diff;
            expval     += sqdiff / softplus(m_isp_global_sigma + *pinpsig++);
        }
    }
    else {
        real global_sigma = softplus(m_isp_global_sigma);
        for (int i=0, n=x1.size() ; i<n ; ++i) {
            real diff   = *px1++ - *px2++;
            real sqdiff = diff * diff;
            expval     += sqdiff / global_sigma;
        }
    }

    // Gate by Kronecker term
    return sf * exp(-0.5 * expval) * gating_term;
}

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void PLearn::SquaredExponentialARDKernel::evaluate_all_i_x ( const Vec x,
const Vec k_xi_x,
real  squared_norm_of_x = -1,
int  istart = 0 
) const [virtual]

Fill k_xi_x with K(x_i, x), for all i from istart to istart + k_xi_x.length() - 1.

Reimplemented from PLearn::Kernel.

Definition at line 277 of file SquaredExponentialARDKernel.cc.

References x.

{
    evaluateAllIXNV<SquaredExponentialARDKernel>(x, k_xi_x, istart);
}
OptionList & PLearn::SquaredExponentialARDKernel::getOptionList ( ) const [virtual]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

OptionMap & PLearn::SquaredExponentialARDKernel::getOptionMap ( ) const [virtual]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

RemoteMethodMap & PLearn::SquaredExponentialARDKernel::getRemoteMethodMap ( ) const [virtual]

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 82 of file SquaredExponentialARDKernel.cc.

void PLearn::SquaredExponentialARDKernel::makeDeepCopyFromShallowCopy ( CopiesMap copies) [virtual]

Transforms a shallow copy into a deep copy.

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 385 of file SquaredExponentialARDKernel.cc.


Member Data Documentation

Reimplemented from PLearn::ARDBaseKernel.

Definition at line 123 of file SquaredExponentialARDKernel.h.


The documentation for this class was generated from the following files:
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