Depth Memory

David González-Martínez; Philipp Nazari; T. Konstantin Rusch; Shiwei Liu

Home/Archive/arXiv

arXiv · 2610.18809

Depth Memory

David González-Martínez·Philipp Nazari·T. Konstantin Rusch·Shiwei Liu

View original source ↗Read source document ↗Export metadata ↓

Abstract

Residual connections propagate information through a network by additively accumulating layer outputs. Recently proposed alternatives, such as Manifold-Constrained Hyper-Connections (mHC) and Attention Residuals (AttnRes), instead use more expressive depth aggregation mechanisms that improve language modeling performance at scale. In this work, we consider the mixing of representations over depth as a problem of storing and retrieving information across layers. Formalizing this perspective, we introduce Depth Memory (DM), a unified framework that describes the standard pre-layer-norm (PreLN) transformer, mHC, and AttnRes in terms of streams that carry representations through the network. One such stream, for example, could be an explicit memory stream that is queried to enrich the input to every layer, and whose output is subsequently written back into the memory for use by later layers. For AttnRes, this memory stream is a growing cache of explicit layer representations. For mHC, it is an expanded residual stream. Building on this view, we propose Linear Depth Memory (LDM), which formulates information retrieval across depth as a regression problem solved using recursive least squares. The resulting architecture propagates three streams across depth: the standard residual stream, a linear associative memory stream, and an inverse covariance stream. For high-throughput training and inference, we propose a hardware-aware implementation of LDM that reduces peak activation memory and memory traffic. LDM consistently achieves the lowest perplexities among PreLN, mHC and AttnRes baselines at the 400M and 1.5B parameter scales, alongside competitive zero-shot accuracy.

Explore related subjects

cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

David González-Martínez, Philipp Nazari, T. Konstantin Rusch, Shiwei Liu. 2026-10-13. Depth Memory. https://arxiv.org/abs/2610.18809

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Generation of data on discontinuous manifolds via continuous stochastic non-invertible networks

The generation of discontinuous distributions is a difficult task for most known frameworks such as generative autoencoders and generative adversarial networks. Generative non-invertible models are unable to accurately generate such distributions, require long training and often are subject to mode collapse. Variational autoencoders (VAEs), which are based on the idea of keeping the latent space to be Gaussian for the sake of a simple sampling, allow an accurate reconstruction, while they experience significant limitations at generation task. In this work, instead of trying to keep the latent space to be Gaussian, we use a pre-trained contrastive encoder to obtain a clustered latent space. Then, for each cluster, representing a unimodal submanifold, we train a dedicated low complexity network to generate this submanifold from the Gaussian distribution. The proposed framework is based on the information-theoretic formulation of mutual information maximization between the input data and latent space representation. We derive a link between the cost functions and the information-theoretic formulation. We apply our approach to synthetic 2D distributions to demonstrate both reconstruction and generation of discontinuous distributions using continuous stochastic networks.

cs.LG↗

arXiv2026

Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series

We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides calibrated prediction intervals. Unlike the existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the average interval lengths by solving a one-dimensional stochastic control problem (SCP) at each time step. In particular, we use the dynamic programming algorithm to find the optimal policy for the SCP. We prove that BCI achieves long-term coverage under arbitrary distribution shifts and temporal dependence, even with poor multi-step ahead forecasts. When the raw forecast intervals are approximately calibrated, we prove that BCI is near optimal for its surrogate model predictive control problem. We find empirically that BCI avoids uninformative intervals with infinite lengths under our main experimental configuration and generates substantially shorter prediction intervals when the raw forecast intervals are poorly calibrated.

cs.LG↗

arXiv2026

GLL: A Differentiable Graph Learning Layer for Neural Networks

Standard deep learning architectures used for classification generate label predictions with a projection head and softmax activation function. Although successful, these methods fail to leverage the relational information between samples for generating label predictions. In recent works, graph-based learning techniques, namely Laplace learning, have been heuristically combined with neural networks for both supervised and semi-supervised learning (SSL) tasks. However, prior works approximate the gradient of the loss function with respect to the graph learning algorithm or decouple the processes; end-to-end integration with neural networks is not achieved. In this work, we derive backpropagation equations, via the adjoint method, for inclusion of a general family of graph learning layers into a neural network. The resulting method, distinct from graph neural networks, allows us to precisely integrate similarity graph construction and graph Laplacian-based label propagation into a neural network layer, replacing a projection head and softmax activation function for general classification task. Our experimental results demonstrate smooth label transitions across data, improved generalization and robustness to adversarial attacks, and improved training dynamics compared to a standard softmax-based approach.

cs.LG↗