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/* MLPACK 0.2
*
* Copyright (c) 2008, 2009 Alexander Gray,
* Garry Boyer,
* Ryan Riegel,
* Nikolaos Vasiloglou,
* Dongryeol Lee,
* Chip Mappus,
* Nishant Mehta,
* Hua Ouyang,
* Parikshit Ram,
* Long Tran,
* Wee Chin Wong
*
* Copyright (c) 2008, 2009 Georgia Institute of Technology
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License as
* published by the Free Software Foundation; either version 2 of the
* License, or (at your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
* 02110-1301, USA.
*/
/**
* @file lin_alg.h
*
* Linear algebra utilities
*
* @author Nishant Mehta
*/
#ifndef LIN_ALG_H
#define LIN_ALG_H
#include "fastlib/fastlib.h"
#define max_rand_i 100000
/**
* Linear algebra utilities.
*
* This includes, among other things, Map, Sum, Addition, Subtraction,
* Multiplication, Hadamard product (entry-wise multiplication), Whitening,
* Random vectors on the unit sphere, Random uniform matrices, Random
* normal matrices, creating a Submatrix that is a slice of selected columns of
* a matrix, Block matrix construction from a base matrix
* Note that the __private is temporary until this code is merged into a larger
* namespace of linear algebra utilities
*/
namespace linalg__private {
/**
* Save the matrix to a file so that rows in the matrix correspond to rows in
* the file: This just means call data::Save() on the transpose of the matrix
*/
void SaveCorrectly(const char *filename, Matrix a) {
Matrix a_transpose;
la::TransposeInit(a, &a_transpose);
data::Save(filename, a_transpose);
}
/**
* Returns \f$ x^{arg} \f$
*/
double ExpArg(double x, double arg) {
return exp(x * arg);
}
/**
* Returns \f$ 1 / x \f$
*/
double Inv(double x, double arg) {
return 1 / x;
}
/**
* Returns \f$ x^2 \f$
*/
double Square(double x, double arg) {
return x * x;
}
/**
* Returns \f$ arg (x^2) \f$
*/
double SquareArg(double x, double arg) {
return arg * x * x;
}
/**
* Returns \f$ \tanh(arg x) \f$
*/
double TanhArg(double x, double arg) {
return tanh(arg * x);
}
/**
* Returns \f$ arg x \f$
*/
double Times(double x, double arg) {
return arg * x;
}
/**
* Returns \f$ x + arg \f$
*/
double Plus(double x, double arg) {
return x + arg;
}
/**
* Returns \f$ x - arg \f$
*/
double MinusArg(double x, double arg) {
return x - arg;
}
/**
* Returns \f$ arg - x \f$
*/
double ArgMinus(double x, double arg) {
return arg - x;
}
/**
* Inits a n by n diagonal matrix and sets the diagonal entries to value
*/
Matrix* DiagMatrixInit(index_t n, double value, Matrix *diag_matrix) {
diag_matrix -> Init(n, n);
diag_matrix -> SetZero();
for(index_t i = 0; i < n; i++) {
diag_matrix -> set(i, i, value);
}
return diag_matrix;
}
/**
* Inits a n-dimensional column vector and sets all entries to value
*/
Matrix* ColVector(index_t n, double value, Matrix *col_vector) {
col_vector -> Init(n, 1);
col_vector -> SetAll(value);
return col_vector;
}
/**
* Sums over the rows of a M by N matrix and Inits a 1 by N matrix storing
* the sum
* (sum_vector \f$ \gets \sum_i \vec{A_{row i}} \f$)
*/
Matrix* Sum(const Matrix* const A, Matrix *sum_vector) {
index_t n_rows = A -> n_rows();
index_t n_cols = A -> n_cols();
sum_vector -> Init(1, n_cols);
const double *A_col_j;
for(index_t j = 0; j < n_cols; j++) {
A_col_j = A -> GetColumnPtr(j);
double sum = 0;
for(index_t i = 0; i < n_rows; i++) {
sum += A_col_j[i];
}
(*sum_vector).set(0, j, sum);
}
return sum_vector;
}
/**
* Returns the sum of the components of vector v
* (returns \f$ \sum v_i \f$)
*/
double Sum(Vector *v) {
index_t n = v -> length();
double sum = 0;
for(index_t i = 0; i < n; i++) {
sum += (*v)[i];
}
return sum;
}
/**
* Applies function with argument arg to a M by N Matrix and Inits a 1 by N
* matrix to the sum over the transformed rows
* (\f$ \tilde{A}_{i,j} \gets function(A_{i,j}, arg) \f$,
* sum_vector \f$ \gets \sum_i \vec{\tilde{A}_{row i}} \f$)
*/
Matrix* MatrixMapSum(double (*function)(double,double),
double arg,
const Matrix* const A,
Matrix *sum_vector) {
index_t n_rows = A -> n_rows();
index_t n_cols = A -> n_cols();
sum_vector -> Init(1, n_cols);
const double *A_col_j;
for(index_t j = 0; j < n_cols; j++) {
A_col_j = A -> GetColumnPtr(j);
double sum = 0;
for(index_t i = 0; i < n_rows; i++) {
sum += function(A_col_j[i], arg);
}
(*sum_vector).set(0, j, sum);
}
return sum_vector;
}
/**
* Applies function with argument arg to vector v and returns the sum of the
* transformed components
* (sum \f$ \gets \sum_i function(v_i, arg) \f$)
*/
double VectorMapSum(double (*function)(double,double),
double arg,
const Vector* const v) {
index_t n = v -> length();
double sum = 0;
for(index_t i = 0; i < n; i++) {
sum += function((*v)[i], arg);
}
return sum;
}
/**
* Multiplies A and B entry-wise and Inits a matrix to the result
* @pre{ A and B are of equal dimensions }
* (\f$ C \gets A .* B \f$)
*/
Matrix* DotMultiplyInit(const Matrix* const A, const Matrix* const B,
Matrix* C) {
index_t n_rows = A -> n_rows();
index_t n_cols = A -> n_cols();
C -> Init(n_rows, n_cols);
const double *A_col_j;
const double *B_col_j;
double *C_col_j;
for(index_t j = 0; j < n_cols; j++) {
A_col_j = A -> GetColumnPtr(j);
B_col_j = B -> GetColumnPtr(j);
C_col_j = C -> GetColumnPtr(j);
for(index_t i = 0; i < n_rows; i++) {
C_col_j[i] = A_col_j[i] * B_col_j[i];
}
}
return C;
}
/**
* Multiplies A and B entry-wise and overwrites matrix B with the result
* @pre{ A and B are of equal dimensions }
* (\f$ B \gets A \bullet B \f$)
*/
Matrix* DotMultiplyOverwrite(const Matrix* const A, Matrix* const B) {
index_t n_rows = A -> n_rows();
index_t n_cols = A -> n_cols();
const double *A_col_j;
double *B_col_j;
for(index_t j = 0; j < n_cols; j++) {
A_col_j = A -> GetColumnPtr(j);
B_col_j = B -> GetColumnPtr(j);
for(index_t i = 0; i < n_rows; i++) {
B_col_j[i] *= A_col_j[i];
}
}
return B;
}
/**
* Multiplies u and v entry-wise and Inits a vector to the result
* @pre{ u and v are of equal dimensions }
* (\f$ \vec{w} \gets \vec{u} \bullet \vec{v} \f$)
*/
Vector* DotMultiplyInit(const Vector* const u, const Vector* const v,
Vector* w) {
index_t n = u -> length();
(*w).Init(n);
for(index_t i = 0; i < n; i++) {
(*w)[i] = (*u)[i] * (*v)[i];
}
return w;
}
/**
* Multiplies u and v entry-wise and overwrites vector v with the result
* @pre{ u and v are of equal dimensions }
* (\f$ \vec{v} \gets \vec{u} \bullet \vec{v} \f$)
*/
Vector* DotMultiplyOverwrite(const Vector* const u, Vector* const v) {
index_t n = u -> length();
for(index_t i = 0; i < n; i++) {
(*v)[i] *= (*u)[i];
}
return v;
}
/**
* Multiplies A and B (N by M matrices) entry-wise and Inits a 1 by N matrix
* to the sum over the transformed rows of the result
* (\f$ C = A \bullet B \f$, sum_vector \f$ \gets \sum_i \vec{C_{row i}} \f$)
*/
Matrix* DotMultiplySum(const Matrix* const A, const Matrix* const B,
Matrix* sum_vector) {
index_t n_rows = A -> n_rows();
index_t n_cols = A -> n_cols();
sum_vector -> Init(1, n_cols);
const double *A_col_j;
const double *B_col_j;
for(index_t j = 0; j < n_cols; j++) {
A_col_j = A -> GetColumnPtr(j);
B_col_j = B -> GetColumnPtr(j);
double sum = 0;
for(index_t i = 0; i < n_rows; i++) {
sum += A_col_j[i] * B_col_j[i];
}
(*sum_vector).set(0, j, sum);
}
return sum_vector;
}
/**
* Inits the diagonal entries of a N by N diagonal matrix to the
* entries of a 1 by N matrix
*/
Matrix* VectorToDiag(const Matrix* const diag_vector, Matrix* diag_matrix) {
index_t n = diag_vector -> n_cols();
diag_matrix -> Init(n, n);
diag_matrix -> SetZero();
for(index_t i = 0; i < n; i++) {
diag_matrix -> set(i, i, diag_vector -> get(0, i));
}
return diag_matrix;
}
/**
* Inits the diagonal entries of a N by N diagonal matrix to the
* entries of a N-dimensional vector
*/
Matrix* VectorToDiag(const Vector* const diag_vector, Matrix* diag_matrix) {
diag_matrix -> InitDiagonal(*diag_vector);
return diag_matrix;
}
/**
* Inits the components of a vector to the diagonal entries of a square matrix
* @pre{ diag_matrix is square }
*/
Vector* DiagToVector(const Matrix* const diag_matrix, Vector* diag_vector) {
index_t n = diag_matrix -> n_rows();
diag_vector -> Init(n);
for(index_t i = 0; i < n; i++) {
(*diag_vector)[i] = diag_matrix -> get(i, i);
}
return diag_vector;
}
/**
* Sets A to alpha * A
* (\f$ A \gets \alpha * A \f$)
*/
Matrix* Scale(double alpha, Matrix *A) {
la::Scale(alpha, A);
return A;
}
/**
* Sets v to alpha * v
* (\f$ \vec{v} \gets \alpha \vec{v} \f$)
*/
Vector* Scale(double alpha, Vector* v) {
la::Scale(alpha, v);
return v;
}
/**
* Inits a matrix to alpha * A
* (\f$ B \gets \alpha A \f$)
*/
Matrix* ScaleInit(double alpha, const Matrix* const A, Matrix* B) {
la::ScaleInit(alpha, *A, B);
return B;
}
/**
* Inits a vector to alpha * u
* (\f$ \vec{v} \gets \alpha \vec{u} \f$)
*/
Vector* ScaleInit(double alpha, const Vector* const u, Vector* v) {
la::ScaleInit(alpha, *u, v);
return v;
}
/**
* Inits a matrix to A * B
* (\f$ C \gets A B \f$)
*/
Matrix* MulInit(const Matrix* const A, const Matrix* const B,
Matrix* const C) {
la::MulInit(*A, *B, C);
return C;
}
/**
* Inits a vector to A * u
* (\f$ \vec{v} \gets A \vec{u} \f$)
*/
Vector* MulInit(const Matrix* const A, const Vector* const u,
Vector* const v) {
la::MulInit(*A, *u, v);
return v;
}
/**
* Inits a vector to u A or A' u
* (\f$ \vec{u} \gets \vec{u} A \f$ or \f$ \vec{u} \gets A^T \vec{u} \f$)
*/
Vector* MulInit(const Vector* const u, const Matrix* const A,
Vector* const v) {
la::MulInit(*u, *A, v);
return v;
}
/**
* Overwrites a matrix with A * B
* (\f$ C \gets A B \f$)
*/
Matrix* MulOverwrite(const Matrix* const A, const Matrix* const B,
Matrix* const C) {
la::MulOverwrite(*A, *B, C);
return C;
}
/**
* Inits a matrix to A' * B
* (\f$ C \gets A^T B \f$)
*/
Matrix* MulTransAInit(const Matrix* const A, const Matrix* const B,
Matrix* C) {
la::MulTransAInit(*A, *B, C);
return C;
}
/**
* Overwrites a matrix with A' * B
* (\f$ C \gets A^T B \f$)
*/
Matrix* MulTransAOverwrite(const Matrix* const A, const Matrix* const B,
Matrix* const C) {
la::MulTransAOverwrite(*A, *B, C);
return C;
}
/**
* Inits a matrix to A * B'
* (\f$ C \gets A B^T \f$)
*/
Matrix* MulTransBInit(const Matrix* const A, const Matrix* const B,
Matrix* C) {
la::MulTransBInit(*A, *B, C);
return C;
}
/**
* Overwrites a matrix with A * B'
* (\f$ C \gets A B^T \f$)
*/
Matrix* MulTransBOverwrite(const Matrix* const A, Matrix* const B,
Matrix* const C) {
la::MulTransBOverwrite(*A, *B, C);
return C;
}
/**
* Inits a matrix to A - B
* (\f$ C \gets A - B \f$)
*/
Matrix* SubInit(const Matrix* const A, const Matrix* const B, Matrix* C) {
la::SubInit(*B, *A, C);
return C;
}
/**
* Inits a vector to u - v
* (\f$ \vec{w} \gets \vec{u} - \vec{v} \f$)
*/
Vector* SubInit(const Vector* const u, const Vector* const v, Vector* w) {
la::SubInit(*v, *u, w);
return w;
}
/**
* Overwrites a matrix with A - B
* (\f$ C \gets A - B \f$)
*/
Matrix* SubOverwrite(const Matrix* const A, const Matrix* const B,
Matrix* const C) {
la::SubOverwrite(*B, *A, C);
return C;
}
/**
* Sets matrix B to B - A
* (\f$ B \gets B - A \f$)
*/
Matrix* SubFrom(const Matrix* const A, Matrix* const B) {
la::SubFrom(*A, B);
return B;
}
/**
* Sets vector v to v - u
* (\f$ \vec{v} \gets \vec{v} - \vec{u} \f$)
*/
Vector* SubFrom(const Vector* const u, Vector* const v) {
la::SubFrom(*u, v);
return v;
}
/**
* Sets matrix B to B + A
* (\f$ B \gets B + A \f$)
*/
Matrix* AddTo(const Matrix* const A, Matrix* const B) {
la::AddTo(*A, B);
return B;
}
/**
* Sets vector v to v + u
* (\f$ \vec{v} \gets \vec{v} + \vec{u} \f$)
*/
Vector* AddTo(const Vector* const u, Vector* const v) {
la::AddTo(*u, v);
return v;
}
/**
* Sets matrix B to B + alpha * A
* (\f$ B \gets B + \alpha A \f$)
*/
Matrix* AddExpert(double alpha, const Matrix* const A, Matrix* const B) {
la::AddExpert(alpha, *A, B);
return B;
}
/**
* Sets vector v to v + alpha * u
* (\f$ \vec{v} \gets \vec{v} + \alpha \vec{u} \f$)
*/
Vector* AddExpert(double alpha, const Vector* const u, Vector* const v) {
la::AddExpert(alpha, *u, v);
return v;
}
/**
* Applies function with argument arg to matrix A and
* overwrites A with the result
* (\f$ A_{i,j} \gets function(A_{i,j}, arg) \f$)
*/
Matrix* MapOverwrite(double (*function)(double,double),
double arg,
Matrix *A) {
index_t n_rows = A -> n_rows();
index_t n_cols = A -> n_cols();
double *A_col_j;
for(index_t j = 0; j < n_cols; j++) {
A_col_j = A -> GetColumnPtr(j);
for(index_t i = 0; i < n_rows; i++) {
A_col_j[i] = function(A_col_j[i], arg);
}
}
return A;
}
/**
* Applies function with argument arg to vector v and
* overwrites v with the result
* (\f$ v_i \gets function(v_i, arg) \f$)
*/
Vector* MapOverwrite(double (*function)(double,double),
double arg,
Vector* const v) {
index_t n = v -> length();
for(index_t i = 0; i < n; i++) {
(*v)[i] = function((*v)[i], arg);
}
return v;
}
/**
* Inits a matrix to the result of applying function with argument arg to
* a matrix
* (\f$ B_{i,j} \gets function(A_{i,j}, arg) \f$)
*/
Matrix* MapInit(double (*function)(double,double),
double arg,
const Matrix* const A,
Matrix *B) {
index_t n_rows = A -> n_rows();
index_t n_cols = A -> n_cols();
B -> Init(n_rows, n_cols);
const double *A_col_j;
double *B_col_j;
for(index_t j = 0; j < n_cols; j++) {
A_col_j = A -> GetColumnPtr(j);
B_col_j = B -> GetColumnPtr(j);
for(index_t i = 0; i < n_rows; i++) {
B_col_j[i] = function(A_col_j[i], arg);
}
}
return B;
}
/**
* Inits a vector to the result of applying function with argument arg to
* a vector
* (\f$ v_i \gets function(u_i, arg) \f$)
*/
Vector* MapInit(double (*function)(double,double),
double arg,
const Vector* const u,
Vector *v) {
index_t n = u -> length();
v -> Init(n);
for(index_t i = 0; i < n; i++) {
(*v)[i] = function((*u)[i], arg);
}
return v;
}
/**
* Inits a matrix to uniform random entries in [0,1]
*/
void RandMatrix(index_t n_rows, index_t n_cols, Matrix *A) {
A -> Init(n_rows, n_cols);
for(index_t j = 0; j < n_cols; j++) {
for(index_t i = 0; i < n_rows; i++) {
A -> set(i, j, drand48());
}
}
}
/**
* Inits a matrix to the columns of A specified in column_indices
*/
void MakeSubMatrixByColumns(Vector column_indices, Matrix A, Matrix *A_sub) {
index_t num_selected = column_indices.length();
A_sub -> Init(A.n_rows(), num_selected);
for(index_t i = 0; i < num_selected; i++) {
index_t index = (index_t) column_indices[i];
Vector A_col_index_i, A_sub_col_i;
A.MakeColumnVector(index, &A_col_index_i);
A_sub -> MakeColumnVector(i, &A_sub_col_i);
A_sub_col_i.CopyValues(A_col_index_i);
}
}
/**
* Sets a matrix to a centered matrix, where centering is done by subtracting
* the sum over the columns (a column vector) from each column of the matrix
*/
void Center(Matrix X, Matrix* X_centered) {
Vector col_vector_sum;
col_vector_sum.Init(X.n_rows());
col_vector_sum.SetZero();
index_t n = X.n_cols();
for(index_t i = 0; i < n; i++) {
Vector cur_col_vector;
X.MakeColumnVector(i, &cur_col_vector);
la::AddTo(cur_col_vector, &col_vector_sum);
}
la::Scale(1/(double) n, &col_vector_sum);
X_centered -> Copy(X);
for(index_t i = 0; i < n; i++) {
Vector cur_col_vector;
X_centered -> MakeColumnVector(i, &cur_col_vector);
la::SubFrom(col_vector_sum, &cur_col_vector);
}
}
/**
* Whitens a matrix using the singular value decomposition of the covariance
* matrix. Whitening means the covariance matrix of the result is
* the identity matrix
*/
void WhitenUsingSVD(Matrix X, Matrix* X_whitened, Matrix* whitening_matrix) {
Matrix cov_X, U, VT, inv_S_matrix, temp1;
Vector S_vector;
Scale(1 / (double) (X.n_cols() - 1),
MulTransBInit(&X, &X, &cov_X));
la::SVDInit(cov_X, &S_vector, &U, &VT);
index_t d = S_vector.length();
inv_S_matrix.Init(d, d);
inv_S_matrix.SetZero();
for(index_t i = 0; i < d; i++) {
double inv_sqrt_val = 1 / sqrt(S_vector[i]);
inv_S_matrix.set(i, i, inv_sqrt_val);
}
MulTransBInit(MulTransAInit(&VT, &inv_S_matrix, &temp1),
&U,
whitening_matrix);
MulInit(whitening_matrix, &X, X_whitened);
}
/**
* Whitens a matrix using the eigen decomposition of the covariance
* matrix. Whitening means the covariance matrix of the result is
* the identity matrix
*/
void WhitenUsingEig(Matrix X, Matrix* X_whitened, Matrix* whitening_matrix) {
Matrix cov_X, D, D_inv, E;
Vector D_vector;
Scale(1 / (double) (X.n_cols() - 1),
MulTransBInit(&X, &X, &cov_X));
la::EigenvectorsInit(cov_X, &D_vector, &E);
//E.set(0, 1, -E.get(0, 1));
//E.set(1, 1, -E.get(1, 1));
index_t d = D_vector.length();
D.Init(d, d);
D.SetZero();
D_inv.Init(d, d);
D_inv.SetZero();
for(index_t i = 0; i < d; i++) {
double sqrt_val = sqrt(D_vector[i]);
D.set(i, i, sqrt_val);
D_inv.set(i, i, 1 / sqrt_val);
}
la::MulTransBInit(D_inv, E, whitening_matrix);
la::MulInit(*whitening_matrix, X, X_whitened);
}
/**
* Overwrites a dimension-N vector to a random vector on the unit sphere in R^N
*/
void RandVector(Vector &v) {
index_t d = v.length();
v.SetZero();
for(index_t i = 0; i+1 < d; i+=2) {
double a = drand48();
double b = drand48();
double first_term = sqrt(-2 * log(a));
double second_term = 2 * M_PI * b;
v[i] = first_term * cos(second_term);
v[i+1] = first_term * sin(second_term);
}
if((d % 2) == 1) {
v[d - 1] = sqrt(-2 * log(drand48())) * cos(2 * M_PI * drand48());
}
la::Scale(1/sqrt(la::Dot(v, v)), &v);
}
/**
* Inits a matrix to random normally distributed entries from N(0,1)
*/
Matrix* RandNormalInit(index_t d, index_t n, Matrix* A) {
double* A_elements = A -> ptr();
index_t num_elements = d * n;
for(index_t i = 0; i+1 < num_elements; i+=2) {
double a = drand48();
double b = drand48();
double first_term = sqrt(-2 * log(a));
double second_term = 2 * M_PI * b;
A_elements[i] = first_term * cos(second_term);
A_elements[i+1] = first_term * sin(second_term);
}
if((d % 2) == 1) {
A_elements[d - 1] = sqrt(-2 * log(drand48())) * cos(2 * M_PI * drand48());
}
return A;
}
/**
* Inits a matrix to a num_row_reps by num_col_reps block matrix where
* each block is base_matrix
*/
Matrix* RepeatMatrix(index_t num_row_reps, index_t num_col_reps,
Matrix base_matrix, Matrix* new_matrix) {
index_t num_rows = base_matrix.n_rows();
index_t num_cols = base_matrix.n_cols();
new_matrix -> Init(num_rows * num_row_reps, num_cols * num_col_reps);
double* base_elements;
double* new_elements = new_matrix -> ptr();
for(index_t col_rep = 0; col_rep < num_col_reps; col_rep++) {
base_elements = base_matrix.ptr();
for(index_t col_num = 0; col_num < num_cols; col_num++) {
for(index_t row_rep = 0; row_rep < num_row_reps; row_rep++) {
memcpy(new_elements, base_elements, num_rows * sizeof(double));
new_elements += num_rows;
}
base_elements += num_rows;
}
}
return new_matrix;
}
/**
* Orthogonalize W and return the result in W, using Eigen Decomposition
* @pre W and W_old store the same matrix in disjoint memory
*/
void Orthogonalize(const Matrix W_old, Matrix *W) {
Matrix W_squared, W_squared_inv_sqrt;
la::MulTransAInit(W_old, W_old, &W_squared);
Matrix D, E, E_times_D;
Vector D_vector;
la::EigenvectorsInit(W_squared, &D_vector, &E);
D.InitDiagonal(D_vector);
index_t d = D.n_rows();
for(index_t i = 0; i < d; i++) {
D.set(i, i, 1 / sqrt(D.get(i, i)));
}
la::MulInit(E, D, &E_times_D);
la::MulTransBInit(E_times_D, E, &W_squared_inv_sqrt);
// note that up until this point, W == W_old
la::MulOverwrite(W_old, W_squared_inv_sqrt, W);
}
}; /* namespace linalg__private */
#endif /* LIN_ALG_H */
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