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Derive Backpropagation for Matrix-Product Layers

Q: Derive Backpropagation for Matrix-Product Layers

This question evaluates understanding of backpropagation through matrix-product layers, covering matrix calculus, the multivariate chain rule, gradient derivation for individual weight matrices, and related linear-algebra competencies.

Consider a neural network block whose output is produced by multiplying a sequence of trainable weight matrices before applying the result to an input.

Let the trainable matrices be $W_1, W_2, \ldots, W_{i-1}$ . Define the cumulative product

$C_i = W_1 W_2 \cdots W_{i-1}.$

Given an input vector or mini-batch $X$ , the forward pass is

$Z_i = C_i X = W_1 W_2 \cdots W_{i-1} X.$

Assume there is a scalar loss function $\mathcal{L}$ , and that the upstream gradient

$G = \frac{\partial \mathcal{L}}{\partial Z_i}$

is provided by the loss function or by later layers.

Derive the backward pass for this block. Specifically:

Express the gradient with respect to each individual matrix $W_j$ , for every $1 \le j < i$ .
Show how the multivariate chain rule applies to the matrix product.
Ensure the resulting gradient $\frac{\partial \mathcal{L}}{\partial W_j}$ has the same shape as $W_j$ .
Describe an efficient implementation that avoids recomputing the same prefix and suffix matrix products repeatedly.

Consider a neural network block whose output is produced by multiplying a sequence of trainable weight matrices before applying the result to an input.

Let the trainable matrices be $W_1, W_2, \ldots, W_{i-1}$ . Define the cumulative product

$C_i = W_1 W_2 \cdots W_{i-1}.$

Given an input vector or mini-batch $X$ , the forward pass is

$Z_i = C_i X = W_1 W_2 \cdots W_{i-1} X.$

Assume there is a scalar loss function $\mathcal{L}$ , and that the upstream gradient

$G = \frac{\partial \mathcal{L}}{\partial Z_i}$

is provided by the loss function or by later layers.

Derive the backward pass for this block. Specifically:

Express the gradient with respect to each individual matrix $W_j$ , for every $1 \le j < i$ .
Show how the multivariate chain rule applies to the matrix product.
Ensure the resulting gradient $\frac{\partial \mathcal{L}}{\partial W_j}$ has the same shape as $W_j$ .
Describe an efficient implementation that avoids recomputing the same prefix and suffix matrix products repeatedly.

Derive Backpropagation for Matrix-Product Layers

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Derive Backpropagation for Matrix-Product Layers

Quick Overview

Solution

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