mHC: Manifold-Constrained Hyper-Connections

Overview

mHC (Manifold-Constrained Hyper-Connections) stabilizes deep network training by constraining residual mixing matrices to be doubly stochastic. It provides:

• Stable Training: Lower gradient norm variance via doubly stochastic constraints
• Multiple Streams: Hyper-Connections with learnable mixing across residual streams
• Sinkhorn Projection: Log-space Sinkhorn-Knopp algorithm for doubly stochastic projection
• GPT Integration: Pattern for wrapping attention and MLP layers

Two components:

• HyperConnections Module: Core PyTorch module with H_res, H_pre, H_post matrices
• Sinkhorn-Knopp: Log-space projection to doubly stochastic manifold

Quick Reference

Topic	Reference
Core Concepts & Math	Core Concepts
Sinkhorn Algorithm	Sinkhorn-Knopp
HyperConnections Module	Module Implementation
GPT Integration	GPT Integration
Common Pitfalls	Pitfalls

Installation

# Required packages
pip install torch einops numpy

Minimal Example

import torch
import torch.nn as nn
from einops import rearrange, einsum

def sinkhorn_knopp(logits, num_iters=20, tau=0.05):
    log_alpha = logits / tau
    for _ in range(num_iters):
        log_alpha = log_alpha - torch.logsumexp(log_alpha, dim=-1, keepdim=True)
        log_alpha = log_alpha - torch.logsumexp(log_alpha, dim=-2, keepdim=True)
    return torch.exp(log_alpha)

class HyperConnections(nn.Module):
    def __init__(self, num_streams, dim, branch=None, layer_idx=0):
        super().__init__()
        self.num_streams = num_streams
        self.branch = branch

        # Initialize H_res near identity (use small negative for gradient flow)
        init_h_res = torch.full((num_streams, num_streams), -0.1)
        init_h_res.fill_diagonal_(0.0)
        self.H_res_logits = nn.Parameter(init_h_res)

        # H_pre/H_post for depth connections
        init_h_pre = torch.full((1, num_streams), -0.1)
        init_h_pre[0, layer_idx % num_streams] = 0.0
        self.H_pre_logits = nn.Parameter(init_h_pre)
        self.H_post_logits = nn.Parameter(torch.zeros(1, num_streams))

    def forward(self, x):
        s = self.num_streams
        x = rearrange(x, "(b s) t d -> b t s d", s=s)

        h_res = sinkhorn_knopp(self.H_res_logits)
        x_mixed = einsum(h_res, x, "s t, b n s d -> b n t d")

        h_pre = self.H_pre_logits.softmax(dim=-1)
        branch_in = einsum(h_pre, x, "v s, b n s d -> b n v d").squeeze(-2)

        branch_out = self.branch(branch_in) if self.branch else branch_in

        h_post = self.H_post_logits.softmax(dim=-1)
        depth_out = einsum(branch_out, h_post, "b t d, v s -> b t s d")

        output = x_mixed + depth_out
        return rearrange(output, "b t s d -> (b s) t d")

Common Imports

import torch
import torch.nn as nn
import torch.nn.functional as F
from einops import rearrange, einsum, repeat, reduce

When to Use What

Scenario	Approach
Standard residual connection	No mHC needed
Deep networks (>12 layers) with stability issues	Use mHC with num_streams=4
GPT/Transformer training	Wrap both attention and MLP with HyperConnections
Custom Sinkhorn iterations	Adjust num_iters (20 default) and tau (0.05 default)
Memory-constrained training	Reduce num_streams or batch size

External Resources

• mHC Paper: https://arxiv.org/abs/2512.24880
• Hyper-Connections: https://arxiv.org/abs/2409.19606
• Sinkhorn's Theorem: https://en.wikipedia.org/wiki/Sinkhorn%27s_theorem