During joint training, backpropagation allows later layers to reshape earlier representations, a feedback mechanism absent from purely layer-wise learning. We extend Neural Low-Degree Filtering (Neural LoFi), a recently proposed layer-wise spectral surrogate for feature learning, to capture this interaction. Starting from the exact backpropagation gradients, we derive a local coupling between a neuron's incoming weights and its corresponding next-layer column. In a controlled small-initialization regime with smooth activations and fixed surroundings, these dynamics amplify singular modes of a cross-layer correlation operator. Each growing mode pairs an earlier feature with an upper-layer response through shared neuron coefficients. This motivates a backward-coupling algorithm in which upper-layer sensitivities generate sample-dependent effective labels for earlier spectral steps, while corrected features refine upward transmission. A function-space variational principle characterizes the selected feature-response pairs by maximizing the label correlation of their product under kernel-norm constraints. For three-layer networks, we derive self-consistent continuous-time dynamics for the filtered hidden-layer features, the activation kernels, and the network output under the proposed backward coupling. In a hierarchical multi-index model, we establish a feature-recovery separation: backward coupling recovers lower-layer directions invisible to the original forward label-weighted statistic. The linear specialization recovers residual-dependent singular-mode dynamics, while paired modes provide a dynamical mechanism for linked low-rank covariance components consistent with the rainbow-network picture. Experiments with fully connected and convolutional networks on binary CIFAR-10 show improved accuracy over forward Neural LoFi at matched widths.
Figures & tables
Figure 1: Backward coupling and paired spikes. The incoming row (blue) and outgoing column (red) meet at neuron j . Upper-layer sensitivity changes the feature learned below, whose activation changes upward transmission. Growing modes form paired rank-one spikes with shared neuron coefficients. The small matrices illustrate a mode concentrated on neuron j : an incoming row and its matching outgoing column.
Figure 2: Backward coupling unlocks the deep staircase. Performance as a function of the sample exponent α=logn/logd . Left: Test MSE. Center: Overlap with the visible first-level block U . Right: Overlap with the hidden blocks V,W . Both spectral methods recover the anchor, the backward correction also recovers the hidden blocks. The prediction panel additionally compares layer-wise and end-to-end gradient descent.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
z1=Φ2(hU).
Appendix
Algorithm 4 Covariance-row BackLoFi for the deep staircase
Stage
Predicted sample scale
Visible U recovery
d2+ϵ
Upper Q,Z recovery
d2ϵ
Learned backward score
d3
Hidden V,W recovery given the score
d2+ϵ
Corrected upper Forward pass
d2ϵ
Appendix
Table 1: Stagewise effective-dimension scales for the quadratic staircase. The d3 score-selection scale uses the contraction size derived in ( 95 ) and requires this contribution to remain spectrally isolated in the complete cross-layer matrix.
Neural network training dynamics studies how parameters and representations change during optimization and how those changes shape learning, stability, and generalization.
Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training in which hierarchical feature learning becomes an explicit iterative spectral procedure. In this limit, the dynamics at each layer decouple: given the current representation, the next layer selects directions with maximal accessible low-degree correlation to the label. This yields a tractable surrogate mechanism for deep learning, together with a natural kernel-space interpretation. Neural LoFi provides a mathematically explicit framework for studying multi-layer feature learning beyond the lazy regime. It predicts how representations are selected layer by layer, explains how emergence of concepts arises with given sample complexity,and gives a concrete mechanism by which depth progressively constructs new features from old ones through low-degree compositionality. We complement the theory with mechanistic experiments on fully connected and convolutional architectures, showing that Neural LoFi improves over lazy random-feature baselines, recovers meaningful structured filters, and predicts representations aligned with early gradient-descent feature discovery with real datasets.
Yatin Dandi, Matteo Vilucchio, Luca Arnaboldi +2
Information Learning and Physics Laboratory, École Polytechnique Fédérale de Lausanne (EPFL) · Statistical Physics of Computation Laboratory, École Polytechnique Fédérale de Lausanne (EPFL)
The Forward-Forward Algorithm (FFA) replaces backpropagation (BP) with layer-wise local contrastive objectives, eliminating the backward pass and the need to retain intermediate activations, yet suffers a persistent performance gap with BP that worsens with depth. This paper diagnoses two structural failure modes: an optimization floor arising from concurrent local updates; and a geometric collapse of layer representations driven by the local update mechanism. On the optimization side, we prove that the FFA loss satisfies the Polyak--Lojasiewicz inequality at each layer; however, simultaneous layer updates induce inter-layer representation-distribution drift, so each layer optimizes against a moving input distribution and incurs an error floor. On the representational side, the pairwise similarity kernel of layer representations contracts exponentially toward rank one as depth increases, collapsing the diversity of per-layer error signals. This collapse bounds FFA's effective learning capacity, which measures the diversity of gradient information across layers, independently of depth, whereas BP's chain-rule signal preserves per-layer diversity, yielding a capacity that scales with depth.
Backpropagation (BP) is widely viewed as biologically implausible, in part because it requires feedback weights to be the transpose of forward weights for error propagation. Interestingly, when training a network with fixed random feedback weights to circumvent this issue, learning aligns the forward weights with the feedback weights, leading the backpropagated error signal to become an approximation of the standard gradient used by BP. This process, called Feedback Alignment (FA), occurs in MLPs and very shallow CNNs but does not scale well to deeper architectures. In this work, we first investigated differences between BP and FA models, trained on CIFAR10, specifically focusing on the effective rank of the signal. We found that the FA error has a considerably lower rank and hence is constrained to a lower-dimensional subspace compared to BP, limiting exploration of the parameter space. Motivated by this observation, we evaluated two mechanisms for increasing the effective dimensionality of FA: Muon, an optimiser that orthogonalises weight updates; and hidden activity normalisation, which promotes activation orthogonality. Across larger architectures and benchmarks, we find that these methods consistently improve over FA baselines, for example, on CIFAR100 with a Resnet-18, accuracy increases by 9 percentage points. Our results identify low-dimensional gradient dynamics as a key obstacle to scaling FA and suggest that inducing higher-dimensional update geometry is a promising route toward scaling alternatives to backpropagation.