Pnorm-class function

The Pnorm class.

This class represents the vector p-norm. class

Pnorm(x, p = 2, axis = NA_real_, keepdims = FALSE, max_denom = 1024)

## S4 method for signature 'Pnorm'
allow_complex(object)

## S4 method for signature 'Pnorm'
to_numeric(object, values)

## S4 method for signature 'Pnorm'
validate_args(object)

## S4 method for signature 'Pnorm'
sign_from_args(object)

## S4 method for signature 'Pnorm'
is_atom_convex(object)

## S4 method for signature 'Pnorm'
is_atom_concave(object)

## S4 method for signature 'Pnorm'
is_atom_log_log_convex(object)

## S4 method for signature 'Pnorm'
is_atom_log_log_concave(object)

## S4 method for signature 'Pnorm'
is_incr(object, idx)

## S4 method for signature 'Pnorm'
is_decr(object, idx)

## S4 method for signature 'Pnorm'
is_pwl(object)

## S4 method for signature 'Pnorm'
get_data(object)

## S4 method for signature 'Pnorm'
name(x)

## S4 method for signature 'Pnorm'
.domain(object)

## S4 method for signature 'Pnorm'
.grad(object, values)

## S4 method for signature 'Pnorm'
.column_grad(object, value)

Arguments

• x: An Expression representing a vector or matrix.
• p: A number greater than or equal to 1, or equal to positive infinity.
• axis: (Optional) The dimension across which to apply the function: 1 indicates rows, 2 indicates columns, and NA indicates rows and columns. The default is NA.
• keepdims: (Optional) Should dimensions be maintained when applying the atom along an axis? If FALSE, result will be collapsed into an $n x 1$ column vector. The default is FALSE.
• max_denom: (Optional) The maximum denominator considered in forming a rational approximation for $p$. The default is 1024.
• object: A Pnorm object.
• values: A list of numeric values for the arguments
• idx: An index into the atom.
• value: A numeric value

Details

If given a matrix variable, Pnorm will treat it as a vector and compute the p-norm of the concatenated columns.

For $p \geq 1$, the p-norm is given by

with domain $x \in \mathbf{R}^n$. For $p < 1, p\neq 0$, the p-norm is given by

with domain $x \in \mathbf{R}^n_+$.

• Note that the "p-norm" is actually a norm only when $p \geq 1$ or $p = +\infty$. For these cases, it is convex.
• The expression is undefined when $p = 0$.
• Otherwise, when $p \< 1$, the expression is concave, but not a true norm.

Methods (by generic)

• allow_complex(Pnorm): Does the atom handle complex numbers?
• to_numeric(Pnorm): The p-norm of x.
• validate_args(Pnorm): Check that the arguments are valid.
• sign_from_args(Pnorm): The atom is positive.
• is_atom_convex(Pnorm): The atom is convex if $p \geq 1$.
• is_atom_concave(Pnorm): The atom is concave if $p \< 1$.
• is_atom_log_log_convex(Pnorm): Is the atom log-log convex?
• is_atom_log_log_concave(Pnorm): Is the atom log-log concave?
• is_incr(Pnorm): The atom is weakly increasing if $p \< 1$ or $p \> 1$ and x is positive.
• is_decr(Pnorm): The atom is weakly decreasing if $p \> 1$ and x is negative.
• is_pwl(Pnorm): The atom is not piecewise linear unless $p = 1$ or $p = \infty$.
• get_data(Pnorm): Returns list(p, axis).
• name(Pnorm): The name and arguments of the atom.
• .domain(Pnorm): Returns constraints describing the domain of the node
• .grad(Pnorm): Gives the (sub/super)gradient of the atom w.r.t. each variable
• .column_grad(Pnorm): Gives the (sub/super)gradient of the atom w.r.t. each column variable

Slots

• x: An Expression representing a vector or matrix.
• p: A number greater than or equal to 1, or equal to positive infinity.
• max_denom: The maximum denominator considered in forming a rational approximation for $p$.
• axis: (Optional) The dimension across which to apply the function: 1 indicates rows, 2 indicates columns, and NA indicates rows and columns. The default is NA.
• keepdims: (Optional) Should dimensions be maintained when applying the atom along an axis? If FALSE, result will be collapsed into an $n x 1$ column vector. The default is FALSE.
• .approx_error: (Internal) The absolute difference between $p$ and its rational approximation.
• .original_p: (Internal) The original input $p$.