picos.expressions.exp_renyientr¶

Implements Renyi entropy expressions.

Classes

class picos.expressions.exp_renyientr.BaseQuasiEntropy(X, Y, alpha)[source]¶

Bases: Expression

Base class defining a general quasi-relative entropy expression.

__ge__(other)[source]¶: Return a constraint that the expression is lower-bounded.

__init__(X, Y, alpha)[source]¶

Construct an BaseQuasiEntropy.

Parameters

X (AffineExpression) – The affine expression $X$ .
Y (AffineExpression) – The affine expression $Y$ . This should have the same dimensions as $X$ .

__le__(other)[source]¶: Return a constraint that the expression is upper-bounded.

property X¶: The expression $X$ .

property Y¶: The additional expression $Y$ .

property alpha¶: The alpha $\alpha$ .

property iscomplex¶: Whether X and Y are complex expressions or not.

property n¶: Lengths of X and Y.

class picos.expressions.exp_renyientr.BaseRenyiEntropy(X, Y, alpha, u=None)[source]¶

Bases: Expression

Base class used to define a general Renyi entropy expression.

__init__(X, Y, alpha, u=None)[source]¶

Construct an BaseRenyiEntropy.

Parameters

X (AffineExpression) – The affine expression $X$ .
Y (AffineExpression) – The affine expression $Y$ . This should have the same dimensions as $X$ .
alpha (float) – The parameter $\alpha$ .
u (AffineExpression) – An additional scalar affine expression $u$ . If specified, then this defines the perspective of the Renyi entropy.

__le__(other)[source]¶: Return a constraint that the expression is upper-bounded.

property X¶: The expression $X$ .

property Y¶: The additional expression $Y$ .

property alpha¶: The alpha $\alpha$ .

property iscomplex¶: Whether X and Y are complex expressions or not.

property n¶: Lengths of X and Y.

property u¶: The additional expression $u$ .

class picos.expressions.exp_renyientr.QuasiEntropy(X, Y, alpha)[source]¶

Bases: BaseQuasiEntropy

Quasi-relative entropy of an affine expression.

Definition

Let $X$ and $Y$ be $N \times N$ -dimensional symmetric or hermitian matrices. Then this is defined as

$\operatorname{Tr}[ X^\alpha Y^{1-\alpha} ],$

for some $\alpha\in[-1, 2]$ .

Warning

When you pose an upper or lower bound on this expression, then PICOS enforces $X \succeq 0$ and $Y \succeq 0$ through an auxiliary constraint during solution search.

class picos.expressions.exp_renyientr.RenyiEntropy(X, Y, alpha, u=None)[source]¶

Bases: BaseRenyiEntropy

Renyi entropy of an affine expression.

Definition

Let $X$ and $Y$ be $N \times N$ -dimensional symmetric or hermitian matrices. Then this is defined as

$\frac{1}{\alpha-1}\log(\operatorname{Tr}[ X^\alpha Y^{1-\alpha} ]),$

for some $\alpha\in[0, 1)$ .

Warning

When you pose an upper or lower bound on this expression, then PICOS enforces $X \succeq 0$ and $Y \succeq 0$ through an auxiliary constraint during solution search.

class picos.expressions.exp_renyientr.SandQuasiEntropy(X, Y, alpha)[source]¶

Bases: BaseQuasiEntropy

Sandwiched quasi-relative entropy of an affine expression.

Definition

Let $X$ and $Y$ be $N \times N$ -dimensional symmetric or hermitian matrices. Then this is defined as

$\operatorname{Tr}[ (Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}})^\alpha ],$

for some $\alpha\in[1/2, 2]$ .

Warning

When you pose an upper or lower bound on this expression, then PICOS enforces $X \succeq 0$ and $Y \succeq 0$ through an auxiliary constraint during solution search.

class picos.expressions.exp_renyientr.SandRenyiEntropy(X, Y, alpha, u=None)[source]¶

Bases: BaseRenyiEntropy

Sandwiched Renyi entropy of an affine expression.

Definition

Let $X$ and $Y$ be $N \times N$ -dimensional symmetric or hermitian matrices. Then this is defined as

$\frac{1}{\alpha-1}\log(\operatorname{Tr}[ (Y^{\frac{1-\alpha}{2\alpha}} X Y^{\frac{1-\alpha}{2\alpha}})^\alpha ]),$

for some $\alpha\in[1/2, 1)$ .

Warning

When you pose an upper or lower bound on this expression, then PICOS enforces $X \succeq 0$ and $Y \succeq 0$ through an auxiliary constraint during solution search.

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