HiResNets: Native Full-HD Video Recognition with Foveal Residual Streams

Much of the recent progress in image and video recognition has come at the cost of memory: larger models, increased resolution, and longer temporal contexts. An inevitable component is the quadratic (or larger) growth of memory and compute based on image resolution, which is a property of the grid sampling used in convolutional networks and vision transformers. In this work we study residual networks whose convolutional blocks have logarithmic-square growth instead, enabling them to process very high-resolution video quickly. The key insight is to use a residual architecture's residual stream as a high-resolution buffer, to which convolutional blocks only read and write via log-polar image warp operations. Layers adaptively focus on different parts of each frame, with very high resolution only near the focus point. A complete high-resolution representation is built up in the residual stream, analogous to eye saccades creating a complete picture in biological vision, and a theoretical construction is presented that eliminates the quadratic dependency of the residual stream resolution. Experiments demonstrate that our proposed HiResNets learn to foveate around scenes similarly to human vision, and have superior performance in difficult egocentric video recognition tasks, especially egocentric video with small objects and fine-grained recognition.

Quantifying Depth Sufficiency in Residual Neural Networks: A First-Order Criterion

How can we determine whether a trained neural network is already deep enough? We study this under a fixed function-preserving residual-growth protocol specifying insertion locations, residual families, zero-output initializations, and zero-state first-order updates. We define first-order residual depth saturation as the absence of a strict local decrease from every admissible insertion. We prove residual non-degeneracy is necessary and sufficient: additional depth has first-order value exactly when conditional activation gradients have a nonzero projection onto at least one admissible residual tangent space. This boundary is shared by descent-compatible zero-state updates and invariant under regular local reparameterizations preserving that tangent space. Under residual-signal realizability, raw activation-gradient vanishing exactly certifies saturation. Across ResNets, GPT-2-style models, and continued-pretrained Pythia checkpoints, the maximum activation-gradient norm decreases toward a low-signal regime with depth. Function-preserving growth also achieves converged performance competitive with training from scratch. These results support activation-gradient magnitude as a conservative diagnostic of the remaining empirical first-order value of residual depth.

Backpropagation as a Nilpotent Linear System

Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs. We present a global operator theory of the \emph{F-adjoint} framework, which reformulates the layerwise backward recursion of an $L$-depth feedforward network into a single linear system $(I-\cB)\Xs=\bG$, where $\bG$ is a source vector. We prove that the global backward operator $\cB$ is strictly block upper-triangular and nilpotent of index at most $L$. This nilpotency guarantees the exact termination of the Neumann series solution after at most $L$ terms, revealing classical backpropagation to be mathematically equivalent to block back-substitution on an upper bidiagonal system. We formalise \emph{F-symmetry} -- the condition in which the backward pass perfectly mirrors the forward pass -- identifying orthogonal weight matrices as canonical examples. Through worked numerical examples, we demonstrate how this operator perspective exposes the single-path collapse of strictly feedforward networks and its breakdown in residual architectures. Finally, we leverage this compositional structure to rigorously derive the mechanics of residual networks (gradient highways) and transfer learning (gradient truncation). This framework elevates backpropagation from an algorithmic recipe to a global nilpotent-operator formulation.

Scaling Adaptive Depth with Norm-Agnostic Residual Networks

Residual architectures are ubiquitous in deep learning, but they suffer from a subtle structural limitation: the norm of the residual stream can grow rapidly with depth. As a result, updates from later layers become small relative to the accumulated residual state. This reduces their impact on the representation and limits the benefits of scaling models in depth. To address this, we introduce NAG, a norm-agnostic residual architecture that separates magnitude from directional information in the residual stream, preserving meaningful layer contributions throughout depth and preventing later updates from being systematically suppressed by residual-norm growth. Importantly, NAG introduces only a negligible number of additional parameters and relies on simple operations that are easily kernel-fusible, preserving training efficiency in practice. We show that this architecture outperforms baseline Transformers, with gains that increase substantially as depth grows, enabling effective training of much deeper models. The norm-agnostic formulation also leads to an interpretable Mixture-of-Depths (MoD) mechanism that adaptively skips both attention and MLP layers. Beyond serving as a post-training accuracy-compute tradeoff, this mechanism can be used as a pretraining-time scaling strategy: under iso-FLOP training, compute saved by reducing per-token forward-pass cost can be reinvested into training on more tokens while keeping the total parameter count and KV-cache budget fixed. In our experiments, moderate Mixture-of-Depths rates of approximately 20%-25% match full-depth baseline performance under equal training compute while substantially reducing the number of executed layer parameters and forward-pass FLOPs. These results identify sparsity in depth as a new scaling axis for fixed-compute training, enabling very deep yet FLOP-efficient models.

Geometric and Spectral Alignment for Deep Neural Network I

Deep residual architectures are modeled as products of near-identity Jacobians. This paper proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors, emphasizing a normalized top-radial Cartan coordinate and fitted power-law chart. Full-rank factors are mapped from $\mathrm{GL}(d)$ to the positive cone by $A\mapsto A^\top A$, then to ordered eigenvalue data. Under Frobenius normalization, exact power-law spectra form a trace-normalized Cartan orbit. This orbit is a Gibbs family on ranks, a Fisher information line, and a Bures--Wasserstein curve with line element $d/4$ times Fisher information. The main rigidity theorem is a slack-aware margin inequality: interface radial amplitude, non-backtracking slack, and signed residual variation control displacement of the fitted Cartan coordinate. In the exact-chart zero-slack case, a depth-$L$ budget gives exponent drift of order $(\log M)/L$; generally, slack and residual increments augment the bound. We separate scalar top-radial from full-Cartan spectral control, which also needs Bures/Hellinger residual variation. We prove approximate-power-law and metric-chart versions, converse lower bounds, Fisher--KL/Bures action estimates, and near-identity expansions for normalized residual chains. Near-identity results verify transport budgets; chart quality remains measurable. Effective rank is a spectral-energy quantile, giving finite-width power-law tail bounds and robust rank-window transition estimates. Empirical static-weight exponent profiles serve as diagnostics; full verification also requires interface budgets, slacks, and residuals for the same operator chain.