MathFunctions.h File Reference

#include <learning/real.h>

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Functions
int	ArgMin (int n, real *x)
int	ArgMax (int n, real *x)
real	SmoothMaxGamma (real f1, real f2, real lambda, real c)
	Approximate max (f1,f2) via a gamma function. More...
real	SmoothMaxPNorm (real f1, real f2, real p)
	Approximate max (f1,f2) via a power function. More...
void	SoftMax (int n, real *Q, real *p, real beta)
void	SoftMin (int n, real *Q, real *p, real beta)
void	Normalise (real *src, real *dst, int n_elements)
	Normalise a vector to a destination vector (low level) More...
real	EuclideanNorm (real *a, real *b, int n)
real	SquareNorm (real *a, real *b, int n)
real	LNorm (real *a, real *b, int n, real p)
real	Sum (real *a, int n)
template<class T >
const T	sign (const T &x)

Function Documentation

ArgMax()

int ArgMax	(	int	n,
		real *	x
	)

Definition at line 36 of file MathFunctions.cpp.

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ArgMin()

int ArgMin	(	int	n,
		real *	x
	)

Definition at line 24 of file MathFunctions.cpp.

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EuclideanNorm()

real EuclideanNorm	(	real *	a,
		real *	b,
		int	n
	)

Definition at line 189 of file MathFunctions.cpp.

LNorm()

real LNorm	(	real *	a,
		real *	b,
		int	n,
		real	p
	)

Definition at line 199 of file MathFunctions.cpp.

Normalise()

void Normalise	(	real *	src,
		real *	dst,
		int	n_elements
	)

Normalise a vector to a destination vector (low level)

src is the source vector. dst is the destination vector. n_elements is the number of elements. As pointers are raw, make sure n_elements is correct. It is safe for src and dst to point at the same vector.

Definition at line 160 of file MathFunctions.cpp.

sign()

template<class T >

const T sign ( const T &  x )

inline

Definition at line 30 of file MathFunctions.h.

SmoothMaxGamma()

real SmoothMaxGamma	(	real	f1,
		real	f2,
		real	lambda,
		real	c
	)

Approximate max (f1,f2) via a gamma function.

We have:

\[ f(x) = \max \{f_1(x), f_2(x)\} = f_1(x) + \max \{0, f_2(x) - f_1(x)\}. \]

We can then approximate the second term with

\[ \max \{0, f_2(x) - f_1(x)\} = \lim_{c \rightarrow \infty} \gamma(f_2(x)-f_1(x), \lambda, c), \]

where \(\lambda \in [0,1], \quad c>0\). The function has the form

\[ \gamma(t,\lambda,c) = \begin{cases} t - \frac{(1-\lambda)^2}{2c} & \textrm{if}~ \frac{1-\lambda}{c} \leq t,\\ \lambda t - \frac{c}{2} t^2 & \textrm{if}~ -\frac{\lambda}{c} \leq t \leq \frac{1-\lambda}{c},\\ -\frac{\lambda^2}{2c} & \text{if}~ t \leq -\frac{\lambda}{c}. \end{cases} \]

The error in the positive region is \((1-\lambda)^2/2c\) and in the negative region it is \(\lambda^2/2c\). The transition region width is \(1/c\).

The advantage of this function is that it is continuously differentiable. Its derivative is

\[ \frac{\partial \gamma(t,\lambda,c)}{\partial t} = \begin{cases} 1 & \textrm{if}~ \frac{1-\lambda}{c} \leq t,\\ \lambda - c t & \textrm{if}~ -\frac{\lambda}{c} \leq t \leq \frac{1-\lambda}{c},\\ 0 & \text{if}~ t \leq -\frac{\lambda}{c}. \end{cases} \]

The error

Definition at line 117 of file MathFunctions.cpp.

SmoothMaxPNorm()

real SmoothMaxPNorm	(	real	f1,
		real	f2,
		real	p
	)

Approximate max (f1,f2) via a power function.

We have:

\[ f(x) = \max \{f_1(x), f_2(x)\} = \lim_{p \rightarrow \infty} (f_1(x)^p, f_2(x)^p)^{1/p}. \]

The advantage of this function is that it is differentiable. Unlike SmoothMaxGamma, the error is not fixed, but it is proportional to the ratio between.

Definition at line 147 of file MathFunctions.cpp.

SoftMax()

void SoftMax	(	int	n,
		real *	Q,
		real *	p,
		real	beta
	)

Definition at line 49 of file MathFunctions.cpp.

SoftMin()

void SoftMin	(	int	n,
		real *	Q,
		real *	p,
		real	beta
	)

Definition at line 63 of file MathFunctions.cpp.

SquareNorm()

real SquareNorm	(	real *	a,
		real *	b,
		int	n
	)

Definition at line 179 of file MathFunctions.cpp.

Sum()

real Sum	(	real *	a,
		int	n
	)

Definition at line 209 of file MathFunctions.cpp.