boundlab.zono.bilinear_elementwise#

boundlab.zono.bilinear_elementwise(A, B)[source]#

Linearize element-wise product of two symbolic expressions.

Both A and B must have the same shape.

The approximation is:

\[A \odot B \approx c_A \odot B + A \odot c_B - c_A \odot c_B + E\]

with element-wise error bound:

\[|E| \le \mathrm{hw}(A) \odot \mathrm{hw}(B)\]
Parameters:

  • A (Expr) – First expression.

  • B (Expr) – Second expression (same shape as A).

Returns:

An expression over-approximating A * B.

Return type:

Expr

Examples

>>> import torch
>>> import boundlab.expr as expr
>>> from boundlab.zono.bilinear import bilinear_elementwise
>>> A = expr.ConstVal(torch.ones(3)) + 0.2 * expr.LpEpsilon([3])
>>> B = expr.ConstVal(torch.zeros(3)) + 0.3 * expr.LpEpsilon([3])
>>> C = bilinear_elementwise(A, B)
>>> C.shape
torch.Size([3])