Backpropagation as a Nilpotent Linear System

• URL: http://arxiv.org/abs/2607.11289v1
• Date: Mon, 13 Jul 2026 09:07:51 GMT
• Title: Backpropagation as a Nilpotent Linear System
• Abstract summary: Backpropagation is the computational engine of deep learning.<n>We present a global operator theory of the emphF-adjoint framework.<n>We show how this operator perspective exposes the single-path collapse of strictly feedforward networks.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• SILAGE: Memory-Efficient, Full-Gradient-Free Nonconvex Optimization for Nested Finite Sums [51.49970814177172]
  Empirical risk on datasets naturally exhibits similarities in $N=nm$ total samples.<n>We provide an analysis that avoids pessimistic convergence analysis.<n>Our results improve existing state-of-the-art regimes.
  arXiv  Detail & Related papers  (2026-06-14T14:11:07Z)
• Can Neural Networks Achieve Optimal Computational-statistical Tradeoff? An Analysis on Single-Index Model [53.6316818897326]
  We propose a gradient-based algorithm for training a two-layer neural network in time.<n>We show that our algorithm learns a sparse representation that strongly aligns with the unknown signal $star$.<n>We extend our approach to the setting where $star$ is $k$-sparse for $k = o(sqrtd)$ by introducing a novel weight technique.
  arXiv  Detail & Related papers  (2026-06-13T09:34:39Z)
• Evolution Strategies at the Hyperscale [57.75314521465674]
  We introduce EGGROLL, an evolution strategies (ES) algorithm designed to scale backprop-free optimization to large population sizes.<n>ES is a set of powerful blackbox optimisation methods that can handle non-differentiable or noisy objectives.<n>EGGROLL overcomes these bottlenecks by generating random matrices $Ain mathbbRmtimes r, Bin mathbbRntimes r$ with $rll min(m,n)$ to form a low-rank matrix perturbation $A Btop$
  arXiv  Detail & Related papers  (2025-11-20T18:56:05Z)
• Bootstrapping as a Morphism: An Arithmetic Geometry Approach to Asymptotically Faster Homomorphic Encryption [0.9179857807576733]
  This paper introduces a new approach to bootstrapping that completely bypasses the traditional circuit evaluation model.<n>We apply the tools of modern arithmetic geometry to reframe the bootstrapping operation as a direct geometric projection.<n>Our work is a complete and provably correct bootstrapping algorithm with a computational complexity of $O(d cdot textpoly(log q))$, where $d$ is the ring dimension and $q$ is the ciphertext.
  arXiv  Detail & Related papers  (2025-09-29T03:39:01Z)
• A Constructive Framework for Nondeterministic Automata via Time-Shared, Depth-Unrolled Feedforward Networks [0.0]
  We present a formal and constructive simulation framework for nondeterministic finite automata (NFAs) using time-shared, depth-unrolled feedforward networks (TS-FFNs)<n>Our formulation symbolically encodes automaton states as binary vectors, transitions as sparse matrix transformations, and nondeterministic branching-including $varepsilon$-closures-as compositions of shared thresholded updates.
  arXiv  Detail & Related papers  (2025-05-30T01:18:35Z)
• Constructive Universal Approximation and Finite Sample Memorization by Narrow Deep ReLU Networks [0.0]
  We show that any dataset with $N$ distinct points in $mathbbRd$ and $M$ output classes can be exactly classified.<n>We also prove a universal approximation theorem in $Lp(Omega; mathbbRm)$ for any bounded domain.<n>Our results offer a unified and interpretable framework connecting controllability, expressivity, and training dynamics in deep neural networks.
  arXiv  Detail & Related papers  (2024-09-10T14:31:21Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• A Unified Scheme of ResNet and Softmax [8.556540804058203]
  We provide a theoretical analysis of the regression problem: $| langle exp(Ax) + A x, bf 1_n rangle-1 ( exp(Ax) + Ax ) This regression problem is a unified scheme that combines softmax regression and ResNet, which has never been done before.
  arXiv  Detail & Related papers  (2023-09-23T21:41:01Z)
• A framework for overparameterized learning [0.0]
  An explanation for the success of deep neural networks is a central question in theoretical machine learning. We propose a framework consisting of a prototype learning problem, which is general enough to cover many popular problems. We then demonstrate that supervised learning, variational autoencoders and training with gradient penalty can be translated to the prototype problem.
  arXiv  Detail & Related papers  (2022-05-26T17:17:46Z)
• Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via GDPA Linearization [59.87663954467815]
  Algorithm unfolding creates an interpretable and parsimonious neural network architecture by implementing each iteration of a model-based algorithm as a neural layer. In this paper, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we unroll a projection-free algorithm for semi-definite programming relaxation (SDR) of a binary graph. Experimental results show that our unrolled network outperformed pure model-based graph classifiers, and achieved comparable performance to pure data-driven networks but using far fewer parameters.
  arXiv  Detail & Related papers  (2021-09-10T07:01:15Z)
• Randomized Exploration for Reinforcement Learning with General Value Function Approximation [122.70803181751135]
  We propose a model-free reinforcement learning algorithm inspired by the popular randomized least squares value iteration (RLSVI) algorithm. Our algorithm drives exploration by simply perturbing the training data with judiciously chosen i.i.d. scalar noises. We complement the theory with an empirical evaluation across known difficult exploration tasks.
  arXiv  Detail & Related papers  (2021-06-15T02:23:07Z)
• Beyond Lazy Training for Over-parameterized Tensor Decomposition [69.4699995828506]
  We show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data. Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
  arXiv  Detail & Related papers  (2020-10-22T00:32:12Z)
• Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography [0.4110409960377149]
  We propose a novel convolutional neural network (CNN) designed for learning pseudodifferential operators ($Psi$DOs) in the context of linear inverse problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN. In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining a wavelet system.
  arXiv  Detail & Related papers  (2020-06-02T14:03:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.