Unlocking the Potential of Similarity Matching: Scalability, Supervision and Pre-training

While effective, the backpropagation (BP) algorithm exhibits limitations in terms of biological plausibility, computational cost, and suitability for online learning. As a result, there has been a growing interest in developing alternative biologically plausible learning approaches that rely on local learning rules. This study focuses on the primarily unsupervised similarity matching (SM) framework, which aligns with observed mechanisms in biological systems and offers online, localized, and biologically plausible algorithms. i) To scale SM to large datasets, we propose an implementation of Convolutional Nonnegative SM using PyTorch. ii) We introduce a localized supervised SM objective reminiscent of canonical correlation analysis, facilitating stacking SM layers. iii) We leverage the PyTorch implementation for pre-training architectures such as LeNet and compare the evaluation of features against BP-trained models. This work combines biologically plausible algorithms with computational efficiency opening multiple avenues for further explorations.

A normative framework for deriving neural networks with multi-compartmental neurons and non-Hebbian plasticity

An established normative approach for understanding the algorithmic basis of neural computation is to derive online algorithms from principled computational objectives and evaluate their compatibility with anatomical and physiological observations. Similarity matching objectives have served as successful starting points for deriving online algorithms that map onto neural networks (NNs) with point neurons and Hebbian/anti-Hebbian plasticity. These NN models account for many anatomical and physiological observations; however, the objectives have limited computational power and the derived NNs do not explain multi-compartmental neuronal structures and non-Hebbian forms of plasticity that are prevalent throughout the brain. In this article, we review and unify recent extensions of the similarity matching approach to address more complex objectives, including a broad range of unsupervised and self-supervised learning tasks that can be formulated as generalized eigenvalue problems or nonnegative matrix factorization problems. Interestingly, the online algorithms derived from these objectives naturally map onto NNs with multi-compartmental neurons and local, non-Hebbian learning rules. Therefore, this unified extension of the similarity matching approach provides a normative framework that facilitates understanding the multi-compartmental neuronal structures and non-Hebbian plasticity found throughout the brain.

Spatial Transformer K-Means

K-means defines one of the most employed centroid-based clustering algorithms with performances tied to the data's embedding. Intricate data embeddings have been designed to push $K$-means performances at the cost of reduced theoretical guarantees and interpretability of the results. Instead, we propose preserving the intrinsic data space and augment K-means with a similarity measure invariant to non-rigid transformations. This enables (i) the reduction of intrinsic nuisances associated with the data, reducing the complexity of the clustering task and increasing performances and producing state-of-the-art results, (ii) clustering in the input space of the data, leading to a fully interpretable clustering algorithm, and (iii) the benefit of convergence guarantees.

A Normative and Biologically Plausible Algorithm for Independent Component Analysis

The brain effortlessly solves blind source separation (BSS) problems, but the algorithm it uses remains elusive. In signal processing, linear BSS problems are often solved by Independent Component Analysis (ICA). To serve as a model of a biological circuit, the ICA neural network (NN) must satisfy at least the following requirements: 1. The algorithm must operate in the online setting where data samples are streamed one at a time, and the NN computes the sources on the fly without storing any significant fraction of the data in memory. 2. The synaptic weight update is local, i.e., it depends only on the biophysical variables present in the vicinity of a synapse. Here, we propose a novel objective function for ICA from which we derive a biologically plausible NN, including both the neural architecture and the synaptic learning rules. Interestingly, our algorithm relies on modulating synaptic plasticity by the total activity of the output neurons. In the brain, this could be accomplished by neuromodulators, extracellular calcium, local field potential, or nitric oxide.

A Similarity-preserving Neural Network Trained on Transformed Images Recapitulates Salient Features of the Fly Motion Detection Circuit

Learning to detect content-independent transformations from data is one of the central problems in biological and artificial intelligence. An example of such problem is unsupervised learning of a visual motion detector from pairs of consecutive video frames. Rao and Ruderman formulated this problem in terms of learning infinitesimal transformation operators (Lie group generators) via minimizing image reconstruction error. Unfortunately, it is difficult to map their model onto a biologically plausible neural network (NN) with local learning rules. Here we propose a biologically plausible model of motion detection. We also adopt the transformation-operator approach but, instead of reconstruction-error minimization, start with a similarity-preserving objective function. An online algorithm that optimizes such an objective function naturally maps onto an NN with biologically plausible learning rules. The trained NN recapitulates major features of the well-studied motion detector in the fly. In particular, it is consistent with the experimental observation that local motion detectors combine information from at least three adjacent pixels, something that contradicts the celebrated Hassenstein-Reichardt model.

A Neural Network with Local Learning Rules for Minor Subspace Analysis

The development of neuromorphic hardware and modeling of biological neural networks requires algorithms with local learning rules. Artificial neural networks using local learning rules to perform principal subspace analysis (PSA) and clustering have recently been derived from principled objective functions. However, no biologically plausible networks exist for minor subspace analysis (MSA), a fundamental signal processing task. MSA extracts the lowest-variance subspace of the input signal covariance matrix. Here, we introduce a novel similarity matching objective for extracting the minor subspace, Minor Subspace Similarity Matching (MSSM). Moreover, we derive an adaptive MSSM algorithm that naturally maps onto a novel neural network with local learning rules and gives numerical results showing that our method converges at a competitive rate.

Interpretable Image Clustering via Diffeomorphism-Aware K-Means

We design an interpretable clustering algorithm aware of the nonlinear structure of image manifolds. Our approach leverages the interpretability of $K$-means applied in the image space while addressing its clustering performance issues. Specifically, we develop a measure of similarity between images and centroids that encompasses a general class of deformations: diffeomorphisms, rendering the clustering invariant to them. Our work leverages the Thin-Plate Spline interpolation technique to efficiently learn diffeomorphisms best characterizing the image manifolds. Extensive numerical simulations show that our approach competes with state-of-the-art methods on various datasets.

A biologically plausible neural network for local supervision in cortical microcircuits

The backpropagation algorithm is an invaluable tool for training artificial neural networks; however, because of a weight sharing requirement, it does not provide a plausible model of brain function. Here, in the context of a two-layer network, we derive an algorithm for training a neural network which avoids this problem by not requiring explicit error computation and backpropagation. Furthermore, our algorithm maps onto a neural network that bears a remarkable resemblance to the connectivity structure and learning rules of the cortex. We find that our algorithm empirically performs comparably to backprop on a number of datasets.

A simple normative network approximates local non-Hebbian learning in the cortex

To guide behavior, the brain extracts relevant features from high-dimensional data streamed by sensory organs. Neuroscience experiments demonstrate that the processing of sensory inputs by cortical neurons is modulated by instructive signals which provide context and task-relevant information. Here, adopting a normative approach, we model these instructive signals as supervisory inputs guiding the projection of the feedforward data. Mathematically, we start with a family of Reduced-Rank Regression (RRR) objective functions which include Reduced Rank (minimum) Mean Square Error (RRMSE) and Canonical Correlation Analysis (CCA), and derive novel offline and online optimization algorithms, which we call Bio-RRR. The online algorithms can be implemented by neural networks whose synaptic learning rules resemble calcium plateau potential dependent plasticity observed in the cortex. We detail how, in our model, the calcium plateau potential can be interpreted as a backpropagating error signal. We demonstrate that, despite relying exclusively on biologically plausible local learning rules, our algorithms perform competitively with existing implementations of RRMSE and CCA.

A biologically plausible neural network for multi-channel Canonical Correlation Analysis

Cortical pyramidal neurons receive inputs from multiple distinct neural populations and integrate these inputs in separate dendritic compartments. We explore the possibility that cortical microcircuits implement Canonical Correlation Analysis (CCA), an unsupervised learning method that projects the inputs onto a common subspace so as to maximize the correlations between the projections. To this end, we seek a multi-channel CCA algorithm that can be implemented in a biologically plausible neural network. For biological plausibility, we require that the network operates in the online setting and its synaptic update rules are local. Starting from a novel CCA objective function, we derive an online optimization algorithm whose optimization steps can be implemented in a single-layer neural network with multi-compartmental neurons and local non-Hebbian learning rules. We also derive an extension of our online CCA algorithm with adaptive output rank and output whitening. Interestingly, the extension maps onto a neural network whose neural architecture and synaptic updates resemble neural circuitry and synaptic plasticity observed experimentally in cortical pyramidal neurons.