Foundations

Affine Projection

EasyRaw PyTorchAbout 10 min2 public examples plus private tests

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Your task

Project the final feature axis of x using the supplied weight, then add bias if present. Return the projected tensor; all leading axes are unchanged.

Function signature

affine_projection(
    x: torch.Tensor,
    weight: torch.Tensor,
    bias: torch.Tensor | None = None,
) -> torch.Tensor

Worked example

Project two vectors

For the first vector, the dot products are $1+4=5$ and $3-2=1$ . Adding the bias gives [5.5, 2]. For the second vector, the dot products are $-1$ and $-3$ , so the result is [-0.5, -2]. Both leading rows are preserved in the (2, 2) output.

import torch

x = torch.tensor([[1.0, 2.0], [-1.0, 0.0]], dtype=torch.float64, requires_grad=True)
weight = torch.tensor([[1.0, 3.0], [2.0, -1.0]], dtype=torch.float64, requires_grad=True)
bias = torch.tensor([0.5, 1.0], dtype=torch.float64, requires_grad=True)

result = affine_projection(x=x, weight=weight, bias=bias)

Expected return

torch.tensor([[5.5, 2.0], [-0.5, -2.0]], dtype=torch.float64)

Parameters

Parameter	Shape and meaning
`x`torch.Tensor	$[\ldots, \mathrm{Cin}]$ Feature vectors whose final axis is projected.
`weight`torch.Tensor	$[\mathrm{Cin}, \mathrm{Cout}]$ Supplied input-channel-first projection matrix.
`bias`torch.Tensor \| NoneDefault `None`	$[\mathrm{Cout}]$ Optional output-channel offset; omission means zero.

Return

torch.Tensor · $[\ldots,C_{out}]$

Return Projected vectors with all leading x axes preserved.

Required behavior

For $x:[\ldots,C_{in}]$ and $W:[C_{in},C_{out}]$ , return $y=xW+b$ . Weight is stored input-channel first, unlike the stored weight of torch.nn.Linear.
bias is optional and has shape (Cout,); omission means no addition. Support arbitrary leading shapes, including empty axes, and noncontiguous tensors.

Why this matters

A projection mixes a vector of features; a lookup selects a learned vector using an integer identity.

Starter code

You write the body of this function in the BioTorch workspace. Your code runs in an isolated PyTorch sandbox; Run checks the public examples and Submit grades the full test suite, including gradients.

import torch


def affine_projection(
    x: torch.Tensor,  # (..., Cin)
    weight: torch.Tensor,  # (Cin, Cout)
    bias: torch.Tensor | None = None,  # (Cout,)
) -> torch.Tensor:
    """Return a tensor with shape (...,C_{out})."""
    # Implement the operation. Inputs and test setup are supplied by BioTorch.
    raise NotImplementedError("Implement affine_projection")

Ready to implement it?

3 progressive hints, a solution walkthrough, and grading that checks values, shapes and gradients are waiting in the workspace.

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Related exercises

Sources and further reading

A primitive exercise, not an end-to-end reproduction of the named models. No affiliation or endorsement is implied.