The Art of Not Forgetting

We introduce CMP (Cognitive Memory Primitive), an architecture that represents inputs as sparse relational codes, stores them in a two-tier competitive memory, and learns entirely through local, gradient-free updates, with no backpropagation anywhere in the network. We use this architecture to test a specific hypothesis: that catastrophic forgetting, usually treated as a training-time defect to be patched with replay or regularization, is instead a structural consequence of how backpropagation assigns credit and that a learning rule that is local and sparse by construction should resist it without a patch. On a controlled domain-incremental protocol across 15 text domains, three-seed replicated, CMP's backward transfer is 15-19x better than a matched-size Transformer trained with online EWC, and the result survives a domain-order control (reported as a range, +0.24 to +0.44, rather than a single figure). We report this alongside a real, substantial accuracy gap versus the Transformer baseline, a null result on a recognized vision benchmark, and a diagnosed, unresolved failure attempting to combine this architecture with a separate mechanism that improves raw accuracy, disclosed because an honest negative result is more useful than an omitted one. The central claim is narrow and falsifiable: local, sparse, non-backpropagation learning measurably resists catastrophic forgetting better than backpropagation with its standard fix, under conditions we state precisely.

On the Sparsity-Storage-Accuracy Tradeoff in Parsimoniously Activated Dictionary Learning

Dictionary learning has long been studied from both optimization and probabilistic perspectives. While formulations with element-wise sparsity regularization (e.g., L1-based sparse coding) admit well-established probabilistic interpretations, many structured variants that impose global constraints lack a clear and tractable generative view. In this paper, we revisit a class of practically effective yet theoretically under-explored dictionary learning methods that impose a simple global regularization on the number of activated dictionary atoms, which we term parsimoniously activated dictionary learning (PADL). We show that PADL admits an equivalent formulation as maximum a posteriori estimation under a structured generative model, with auxiliary latent variables that govern global activation patterns. This formulation allows us to derive generalization guarantees that are difficult to obtain under the original formulation. More importantly, it yields an analytical characterization of the tradeoff between sparsity, storage cost, and reconstruction accuracy, enabling data-driven estimation of optimal hyperparameters. Based on this connection, we develop an efficient and interpretable PADL algorithm that eliminates manual hyperparameter tuning, achieving improved reconstruction performance under comparable sparsity levels on visual benchmarks. We further demonstrate its practical utility in accelerating inference for vision-language models.

Effects of sparsity and superposition on loss in simple autoencoders

One of the major difficulties in the mechanistic interpretability of neural networks is the occurrence of polysemanticity, which suggests that each neuron is typically responsible for multiple different tasks, impeding a clean interpretation of their function. The seminal paper of Elhage et al. (2022) argues that this occurs due to superposition, a phenomenon where the neural network represents distinct features as non-orthogonal directions in a lower-dimensional space, a strategy that allows much greater compression of the data without sacrificing fidelity due to the feature sparsity of input vectors. Elhage et al. (2022) empirically validates these hypotheses in a rather natural and simple autoencoder with sparse inputs. The contribution of the present work is to analyze the mathematical basis for the occurrence and optimality of superposition, while rigorously corroborating some of their findings. In particular, we provide upper and lower bounds for the L2 reconstruction loss, tight in the very sparse regime, for power activation functions. A short list of interesting open problems are also included at the end.

Learning Doubly Sparse Explicitly Conditioned Transforms

Finding convenient spaces in which certain hypotheses regarding an assumed sparse structure of natural signals hold true has become a desirable result in recent research, its implications being reflected in areas such as data compression, noise reduction and feature extraction. While the extensively used analytical transforms, such as DFT or DCT, already provide efficient algorithms and robust sparse representations, they assume a fixed prior about the data, failing to accurately capture the specific structure of more restrictive classes of signals. To address this, the concept of a data-adaptive, learnt transform has been introduced in the literature, allowing for the reduction of a residual term in the transform domain. More recent studies have shown that the condition number serves as a good metric in this context, where the desired outcome alternates between a generalizing tendency and one that achieves minimal approximation error. Motivated by these considerations, we introduce the learning of a structured, explicitly conditioned transform formulated as the product of a fixed canonical matrix and a refining data-adaptive sparse component. This approach seeks to preserve the advantages of fast and stable analytical transforms, while introducing controllable adaptivity to the data. No references that concern this specific formulation have been identified so far, indicating its novelty. The proposed algorithm is motivated within the framework of inexact proximal methods, leveraging a newly derived closed-form projection operator. Empirical observations demonstrate state-of-the-art results on the doubly sparse transform learning problem and comparable performance with its dense variant at significantly lower computational costs and sometimes faster convergence and better avoidance of bad local minima.

Flexible Online Representation Learning Based on Similarity Matching

Sparse high-dimensional representations are conducive to uncovering nontrivial structures in unsupervised exploration of data. Such a representation can deal with the dense connectivity in graphs relevant to community detection problems. However, sparse high-dimensional representations are capable of doing more, including manifold tiling and feature learning. Conventional algorithms optimize in the space of computationally intractable completely positive matrices or relax the problem to the space of doubly nonnegative matrices that scale with sample size in a way rendering them impractical for large data sets. Some of these methods also impose a row sum constraint, such as double stochasticity. Row sum constraints have the added advantage of being shift-invariant, in the context of manifold tiling. Constraints on the row sum of output similarity matrices require nontrivial online learning rules. Addressing these needs, we propose a versatile online biologically plausible learning algorithm capable of learning sparse shift-invariant representations, useful for clustering, manifold tiling, or sparse coding, depending on the data structure.