Doubly Stochastic Stream Mixing Forward Pass (Medium) — Practice with Code Visualizer

In modern deep learning architectures, multi-stream residual connections have emerged as a powerful mechanism for enhancing gradient flow and enabling diverse feature representations. A particularly sophisticated variant introduces constrained mixing matrices that combine information from parallel streams while maintaining mathematical stability during training.

The core innovation involves doubly stochastic mixing matrices—square matrices where every row and every column sums to exactly 1. This constraint ensures that:

Information is neither amplified nor diminished in aggregate during mixing
The transformation preserves the overall "energy" or magnitude distribution across streams
Training remains stable even with many parallel paths

Sinkhorn Normalization Algorithm: To project an arbitrary matrix of raw coefficients into the space of doubly stochastic matrices, we use the Sinkhorn-Knopp algorithm:

Start with the exponentiated raw matrix: M = exp(H_raw)
Repeat for a fixed number of iterations (typically 10):
- Normalize each row to sum to 1: M = M / row_sums(M)
- Normalize each column to sum to 1: M = M / col_sums(M)
After convergence, M approximates a doubly stochastic matrix

Forward Pass Computation: Given n parallel streams of hidden states x with shape (n, d), the mixing operation proceeds as:

Convert raw residual coefficients H_res_raw to doubly stochastic form via Sinkhorn
Compute the mixed residual: output = H_res @ x
(Optional) Add contributions from downstream layers using H_post coefficients

Your Task: Implement the forward pass of this stream mixing mechanism. Given the parallel stream states and raw coefficient matrices, apply the Sinkhorn normalization to create the doubly stochastic mixing matrix, then compute the mixed output.

Implementation Details:

Use 10 iterations of Sinkhorn normalization
Apply element-wise exponential before Sinkhorn iteration
Handle the matrix multiplication for stream mixing
Round final output values to 2 decimal places

With H_res_raw containing all zeros, exp(0) = 1 creates a 2×2 matrix of ones.

Sinkhorn normalization on this uniform matrix:

After row normalization: [[0.5, 0.5], [0.5, 0.5]]
After column normalization: [[0.5, 0.5], [0.5, 0.5]]
The matrix is already doubly stochastic!

Mixing computation: H_res @ x = [[0.5, 0.5], [0.5, 0.5]] @ [[1, 0], [0, 1]] = [[0.5×1 + 0.5×0, 0.5×0 + 0.5×1], [0.5×1 + 0.5×0, 0.5×0 + 0.5×1]] = [[0.5, 0.5], [0.5, 0.5]]

Each output stream becomes a uniform average of the two input streams.

With H_res_raw = [[1, 0], [0, 1]], the exponentiated matrix is: exp([[1, 0], [0, 1]]) = [[e, 1], [1, e]] ≈ [[2.718, 1], [1, 2.718]]

After Sinkhorn iterations, this converges to approximately: H_res ≈ [[0.73, 0.27], [0.27, 0.73]]

The diagonal dominance in H_res_raw translates to higher weights on the diagonal after projection.

Mixing: H_res @ x ≈ [[0.73×1 + 0.27×3, 0.73×2 + 0.27×4], [0.27×1 + 0.73×3, 0.27×2 + 0.73×4]] ≈ [[1.54, 2.54], [2.46, 3.46]]

Each stream is a weighted combination favoring its original position.

With 3 streams and all-zero H_res_raw, exp(0) = 1 gives a 3×3 matrix of ones.

Sinkhorn normalization converges to a uniform doubly stochastic matrix: H_res = [[1/3, 1/3, 1/3], [1/3, 1/3, 1/3], [1/3, 1/3, 1/3]]

Since x is the 3×3 identity matrix: H_res @ I = H_res ≈ [[0.33, 0.33, 0.33], [0.33, 0.33, 0.33], [0.33, 0.33, 0.33]]

Each output stream receives equal contributions from all input streams.