Implement Bilinear Form with Einsum

hardPROEinsum OperationsPyTorch Advanced

Problem Statement

Use einsum to compute a batched bilinear form x^T A y.

Background

A bilinear form computes x^T A y where x and y are vectors and A is a matrix. With einsum, you can express this contraction over multiple indices in a single call.

Your Task

The starter code creates x (3x4), A (4x4), and y (3x4). Use einsum to compute the batched bilinear form x^T A y, producing one scalar per batch element.

Output Format

Returns a dictionary with "results" (one scalar per batch), "shape", and "first_result".

Example

Input

None

Output

{'results': [1.0, 6.0, 136.0], 'shape': [3], 'first_result': 1.0}

Reasoning

Inputs: $x$ and $y$ are batches of three 4-dimensional vectors (row $b$ is $x_b$ or $y_b$ ), and $A$ is a single $4 \times 4$ matrix with entries $1$ to $16$ , shared by every batch element.
Math: for each $b$ , the bilinear form is $x_b^T A y_b = \sum_i \sum_j x_{bi} A_{ij} y_{bj}$ . Both $i$ and $j$ are summed away; only $b$ survives.
$b=0$ : $x_0 = y_0 = e_1$ (the first unit vector), so only the term $A_{11} = 1$ survives, giving $1.0$ .
$b=1$ : $x_1 = y_1 = e_2$ picks out $A_{22} = 6$ , giving $6.0$ .
$b=2$ : all-ones vectors keep every term, so the result is the sum of all entries of $A$ : row sums $10 + 26 + 42 + 58 = 136$ .
Output: one scalar per batch element, so results is [1.0, 6.0, 136.0], shape is [3], and first_result is 1.0. Notice that with unit vectors $e_i^T A e_j$ simply reads entry $(i, j)$ of $A$ .

Constraints

Use 'bi,ij,bj->b' einsum notation
Batched computation
Fixed input values for determinism

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Background Knowledge

The problem involves computing a bilinear form, which is a mathematical operation that takes two vectors and a matrix as input and produces a scalar output. In this case, we're working with batched vectors, meaning we have multiple vectors of the same dimension, and we want to compute the bilinear form for each batch. The bilinear form is defined as $x^T A y$ , where $x$ and $y$ are vectors and $A$ is a matrix. This operation can be thought of as a weighted dot product between $x$ and $y$ , where the weights are given by the matrix $A$ .

In the context of PyTorch, we're using the torch.einsum function to compute the bilinear form. Einsum is a powerful function that allows us to specify complex tensor operations using a concise notation. It's particularly useful for batched operations, where we need to perform the same operation on multiple tensors. The torch.einsum function takes a string argument that specifies the operation to be performed, as well as the input tensors. In this case, we're using the string 'bi,ij,bj->b' to specify the bilinear form operation.

The bilinear form has many applications in machine learning and deep learning, including neural networks, attention mechanisms, and kernel methods. Understanding how to compute the bilinear form efficiently is crucial for many of these applications. In this problem, we're working with fixed values for $x$ , $A$ , and $y$ , but in practice, these values would typically be learned during training or computed on the fly.

Algorithm/Approach

The general approach to solving this problem involves using the torch.einsum function to compute the bilinear form. We'll need to specify the input tensors $x$ , $A$ , and $y$ , as well as the output tensor shape. The torch.einsum function will then perform the necessary computations to produce the output tensor. We'll also need to extract the results from the output tensor and return them in the required format.

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solution.py

Python 3.13

Test results