scico.loss¶

Loss function classes.

Classes

`Loss`(y[, A, f, scale])	Generic loss function.
`PoissonLoss`(y[, A, scale])	Poisson negative log likelihood loss.
`SquaredL2AbsLoss`(y[, A, scale, W])	Weighted squared \(\ell_2\) with absolute value loss.
`SquaredL2Loss`(y[, A, scale, W, prox_kwargs])	Weighted squared \(\ell_2\) loss.
`SquaredL2SquaredAbsLoss`(y[, A, scale, W])	Weighted squared \(\ell_2\) with squared absolute value loss.

class scico.loss.Loss(y, A=None, f=None, scale=1.0)[source]¶

Bases: Functional

Generic loss function.

Generic loss function

\[\alpha f(\mb{y}, A(\mb{x})) \;,\]

where \(\alpha\) is the scaling parameter and \(f(\cdot)\) is the loss functional.

Parameters:

y (Union[Array, BlockArray]) – Measurement.
A (Union[Callable, Operator, None]) – Forward operator. Defaults to None, in which case self.A is a Identity with input shape and dtype determined by the shape and dtype of y.
f (Optional[Functional]) – Functional \(f\). If defined, the loss function is \(\alpha f(\mb{y} - A(\mb{x}))\). If None, then __call__ and prox (where appropriate) must be defined in a derived class.
scale (float) – Scaling parameter. Default: 1.0.

__call__(x)[source]¶

Evaluate this loss at point \(\mb{x}\).

Parameters:: x (Union[Array, BlockArray]) – Point at which to evaluate loss.
Return type:: float
Returns:: Result of evaluating the loss at x.

prox(v, lam=1, **kwargs)[source]¶

Scaled proximal operator of loss function.

Evaluate scaled proximal operator of this loss function, with scaling \(\lambda\) = lam and evaluated at point \(\mb{v}\) = v. If prox is not defined in a derived class, and if operator \(A\) is the identity operator, then the proximal operator is computed using the proximal operator of functional \(l\), via Theorem 6.11 in [7].

Parameters:

v (Union[Array, BlockArray]) – Point at which to evaluate prox function.
lam (float) – Proximal parameter \(\lambda\).
**kwargs – Additional arguments that may be used by derived classes. These include x0, an initial guess for the minimizer in the defintion of \(\mathrm{prox}\).

Return type:

Union[Array, BlockArray]

Returns:

Result of evaluating the scaled proximal operator at v.

set_scale(new_scale)[source]¶: Update the scale attribute.

class scico.loss.SquaredL2Loss(y, A=None, scale=0.5, W=None, prox_kwargs=None)[source]¶

Bases: Loss

Weighted squared \(\ell_2\) loss.

Weighted squared \(\ell_2\) loss

\[\alpha \norm{\mb{y} - A(\mb{x})}_W^2 = \alpha \left(\mb{y} - A(\mb{x})\right)^T W \left(\mb{y} - A(\mb{x})\right) \;,\]

where \(\alpha\) is the scaling parameter and \(W\) is an instance of scico.linop.Diagonal. If \(W\) is None, the weighting is an identity operator, giving an unweighted squared \(\ell_2\) loss.

Parameters:

y (Union[Array, BlockArray]) – Measurement.
A (Union[Callable, Operator, None]) – Forward operator. If None, defaults to Identity.
scale (float) – Scaling parameter.
W (Optional[Diagonal]) – Weighting diagonal operator. Must be non-negative. If None, defaults to Identity.

__call__(x)[source]¶

Evaluate this loss at point \(\mb{x}\).

Parameters:: x (Union[Array, BlockArray]) – Point at which to evaluate loss.
Return type:: float
Returns:: Result of evaluating the loss at x.

prox(v, lam=1.0, **kwargs)[source]¶

Scaled proximal operator of loss function.

Parameters:

v (Union[Array, BlockArray]) – Point at which to evaluate prox function.
lam (float) – Proximal parameter \(\lambda\).
**kwargs – Additional arguments that may be used by derived classes. These include x0, an initial guess for the minimizer in the defintion of \(\mathrm{prox}\).

Return type:

Union[Array, BlockArray]

Returns:

Result of evaluating the scaled proximal operator at v.

property hessian: LinearOperator¶

Compute the Hessian of linear operator A.

If self.A is a scico.linop.LinearOperator, returns a scico.linop.LinearOperator corresponding to the Hessian \(2 \alpha \mathrm{A^H W A}\). Otherwise not implemented.

class scico.loss.PoissonLoss(y, A=None, scale=0.5)[source]¶

Bases: Loss

Poisson negative log likelihood loss.

Poisson negative log likelihood loss

\[\alpha \left( \sum_i [A(x)]_i - y_i \log\left( [A(x)]_i \right) + \log(y_i!) \right) \;,\]

where \(\alpha\) is the scaling parameter.

Parameters:

y (Union[Array, BlockArray]) – Measurement.
A (Union[Callable, Operator, None]) – Forward operator. Defaults to None, in which case self.A is a Identity.
scale (float) – Scaling parameter. Default: 0.5.

const¶: Constant term, \(\ln(y!)\), in Poisson log likehood.

__call__(x)[source]¶

Evaluate this loss at point \(\mb{x}\).

Parameters:: x (Union[Array, BlockArray]) – Point at which to evaluate loss.
Return type:: float
Returns:: Result of evaluating the loss at x.

class scico.loss.SquaredL2AbsLoss(y, A=None, scale=0.5, W=None)[source]¶

Bases: Loss

Weighted squared \(\ell_2\) with absolute value loss.

Weighted squared \(\ell_2\) with absolute value loss

\[\alpha \norm{\mb{y} - | A(\mb{x}) |\,}_W^2 = \alpha \left(\mb{y} - | A(\mb{x}) |\right)^T W \left(\mb{y} - | A(\mb{x}) |\right) \;,\]

where \(\alpha\) is the scaling parameter and \(W\) is an instance of scico.linop.Diagonal.

Proximal operator prox is implemented when \(A\) is an instance of scico.linop.Identity. This is not proximal operator according to the strict definition since the loss function is non-convex (Sec. 3) [52].

Parameters:

y (Union[Array, BlockArray]) – Measurement.
A (Union[Callable, Operator, None]) – Forward operator. If None, defaults to Identity.
scale (float) – Scaling parameter.
W (Optional[Diagonal]) – Weighting diagonal operator. Must be non-negative. If None, defaults to Identity.

__call__(x)[source]¶

Evaluate this loss at point \(\mb{x}\).

Parameters:: x (Union[Array, BlockArray]) – Point at which to evaluate loss.
Return type:: float
Returns:: Result of evaluating the loss at x.

prox(v, lam=1.0, **kwargs)[source]¶

Scaled proximal operator of loss function.

Parameters:

v (Union[Array, BlockArray]) – Point at which to evaluate prox function.
lam (float) – Proximal parameter \(\lambda\).
**kwargs – Additional arguments that may be used by derived classes. These include x0, an initial guess for the minimizer in the defintion of \(\mathrm{prox}\).

Return type:

Union[Array, BlockArray]

Returns:

Result of evaluating the scaled proximal operator at v.

class scico.loss.SquaredL2SquaredAbsLoss(y, A=None, scale=0.5, W=None)[source]¶

Bases: Loss

Weighted squared \(\ell_2\) with squared absolute value loss.

Weighted squared \(\ell_2\) with squared absolute value loss

\[\alpha \norm{\mb{y} - | A(\mb{x}) |^2 \,}_W^2 = \alpha \left(\mb{y} - | A(\mb{x}) |^2 \right)^T W \left(\mb{y} - | A(\mb{x}) |^2 \right) \;,\]

where \(\alpha\) is the scaling parameter and \(W\) is an instance of scico.linop.Diagonal.

Parameters:

y (Union[Array, BlockArray]) – Measurement.
A (Union[Callable, Operator, None]) – Forward operator. If None, defaults to Identity.
scale (float) – Scaling parameter.
W (Optional[Diagonal]) – Weighting diagonal operator. Must be non-negative. If None, defaults to Identity.

__call__(x)[source]¶

Evaluate this loss at point \(\mb{x}\).

Parameters:: x (Union[Array, BlockArray]) – Point at which to evaluate loss.
Return type:: float
Returns:: Result of evaluating the loss at x.

prox(v, lam=1.0, **kwargs)[source]¶

Scaled proximal operator of loss function.

Parameters:

v (Union[Array, BlockArray]) – Point at which to evaluate prox function.
lam (float) – Proximal parameter \(\lambda\).
**kwargs – Additional arguments that may be used by derived classes. These include x0, an initial guess for the minimizer in the defintion of \(\mathrm{prox}\).

Return type:

Union[Array, BlockArray]

Returns:

Result of evaluating the scaled proximal operator at v.