Geometry is All You Need: A Unified Taxonomy of Matrix and Tensor Factorization for Compression of Generative Language Models

Matrix and tensor-guided parametrization for Natural Language Processing (NLP) models is fundamentally useful for the improvement of the model's systematic efficiency. However, the internal links between these two algebra structures and language model parametrization are poorly understood. Also, the existing matrix and tensor research is math-heavy and far away from machine learning (ML) and NLP research concepts. These two issues result in the recent progress on matrices and tensors for model parametrization being more like a loose collection of separate components from matrix/tensor and NLP studies, rather than a well-structured unified approach, further hindering algorithm design. To this end, we propose a unified taxonomy, which bridges the matrix/tensor compression approaches and model compression concepts in ML and NLP research. Namely, we adopt an elementary concept in linear algebra, that of a subspace, which is also the core concept in geometric algebra, to reformulate the matrix/tensor and ML/NLP concepts (e.g. attention mechanism) under one umbrella. In this way, based on our subspace formalization, typical matrix and tensor decomposition algorithms can be interpreted as geometric transformations. Finally, we revisit recent literature on matrix- or tensor-guided language model compression, rephrase and compare their core ideas, and then point out the current research gap and potential solutions.

In-ear ECG Signal Enhancement with Denoising Convolutional Autoencoders

The cardiac dipole has been shown to propagate to the ears, now a common site for consumer wearable electronics, enabling the recording of electrocardiogram (ECG) signals. However, in-ear ECG recordings often suffer from significant noise due to their small amplitude and the presence of other physiological signals, such as electroencephalogram (EEG), which complicates the extraction of cardiovascular features. This study addresses this issue by developing a denoising convolutional autoencoder (DCAE) to enhance ECG information from in-ear recordings, producing cleaner ECG outputs. The model is evaluated using a dataset of in-ear ECGs and corresponding clean Lead I ECGs from 45 healthy participants. The results demonstrate a substantial improvement in signal-to-noise ratio (SNR), with a median increase of 5.9 dB. Additionally, the model significantly improved heart rate estimation accuracy, reducing the mean absolute error by almost 70% and increasing R-peak detection precision to a median value of 90%. We also trained and validated the model using a synthetic dataset, generated from real ECG signals, including abnormal cardiac morphologies, corrupted by pink noise. The results obtained show effective removal of noise sources with clinically plausible waveform reconstruction ability.

Interpretable Pre-Trained Transformers for Heart Time-Series Data

Decoder-only transformers are the backbone of the popular generative pre-trained transformer (GPT) series of large language models. In this work, we apply the same framework to periodic heart time-series data to create two pre-trained general purpose cardiac models, namely PPG-PT and ECG-PT. We demonstrate that both such pre-trained models are fully interpretable. This is achieved firstly through aggregate attention maps which show that the model focuses on similar points in previous cardiac cycles in order to make predictions and gradually broadens its attention in deeper layers. Next, tokens with the same value, that occur at different distinct points in the ECG and PPG cycle, form separate clusters in high dimensional space based on their phase as they propagate through the transformer blocks. Finally, we highlight that individual attention heads respond to specific physiologically relevent features, such as the dicrotic notch in PPG and the P-wave in ECG. It is also demonstrated that these pre-trained models can be easily fine-tuned for tasks such as classification of atrial fibrillation. In this specific example, the fine-tuning took 11 minutes of computer time, and achieved a leave-one-subject-out AUCs of 0.99 and 0.93 for ECG and PPG respectively. Importantly, these fine-tuned models are also fully explainable, with attention shifting to regions in the context that are strongly indicative of atrial fibrillation.

Demystifying the Hypercomplex: Inductive Biases in Hypercomplex Deep Learning

Hypercomplex algebras have recently been gaining prominence in the field of deep learning owing to the advantages of their division algebras over real vector spaces and their superior results when dealing with multidimensional signals in real-world 3D and 4D paradigms. This paper provides a foundational framework that serves as a roadmap for understanding why hypercomplex deep learning methods are so successful and how their potential can be exploited. Such a theoretical framework is described in terms of inductive bias, i.e., a collection of assumptions, properties, and constraints that are built into training algorithms to guide their learning process toward more efficient and accurate solutions. We show that it is possible to derive specific inductive biases in the hypercomplex domains, which extend complex numbers to encompass diverse numbers and data structures. These biases prove effective in managing the distinctive properties of these domains, as well as the complex structures of multidimensional and multimodal signals. This novel perspective for hypercomplex deep learning promises to both demystify this class of methods and clarify their potential, under a unifying framework, and in this way promotes hypercomplex models as viable alternatives to traditional real-valued deep learning for multidimensional signal processing.

Quaternion recurrent neural network with real-time recurrent learning and maximum correntropy criterion

We develop a robust quaternion recurrent neural network (QRNN) for real-time processing of 3D and 4D data with outliers. This is achieved by combining the real-time recurrent learning (RTRL) algorithm and the maximum correntropy criterion (MCC) as a loss function. While both the mean square error and maximum correntropy criterion are viable cost functions, it is shown that the non-quadratic maximum correntropy loss function is less sensitive to outliers, making it suitable for applications with multidimensional noisy or uncertain data. Both algorithms are derived based on the novel generalised HR (GHR) calculus, which allows for the differentiation of real functions of quaternion variables and offers the product and chain rules, thus enabling elegant and compact derivations. Simulation results in the context of motion prediction of chest internal markers for lung cancer radiotherapy, which includes regular and irregular breathing sequences, support the analysis.

Widely Linear Matched Filter: A Lynchpin towards the Interpretability of Complex-valued CNNs

A recent study on the interpretability of real-valued convolutional neural networks (CNNs) {Stankovic_Mandic_2023CNN} has revealed a direct and physically meaningful link with the task of finding features in data through matched filters. However, applying this paradigm to illuminate the interpretability of complex-valued CNNs meets a formidable obstacle: the extension of matched filtering to a general class of noncircular complex-valued data, referred to here as the widely linear matched filter (WLMF), has been only implicit in the literature. To this end, to establish the interpretability of the operation of complex-valued CNNs, we introduce a general WLMF paradigm, provide its solution and undertake analysis of its performance. For rigor, our WLMF solution is derived without imposing any assumption on the probability density of noise. The theoretical advantages of the WLMF over its standard strictly linear counterpart (SLMF) are provided in terms of their output signal-to-noise-ratios (SNRs), with WLMF consistently exhibiting enhanced SNR. Moreover, the lower bound on the SNR gain of WLMF is derived, together with condition to attain this bound. This serves to revisit the convolution-activation-pooling chain in complex-valued CNNs through the lens of matched filtering, which reveals the potential of WLMFs to provide physical interpretability and enhance explainability of general complex-valued CNNs. Simulations demonstrate the agreement between the theoretical and numerical results.

The HR-Calculus: Enabling Information Processing with Quaternion Algebra

From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.

TensorGPT: Efficient Compression of the Embedding Layer in LLMs based on the Tensor-Train Decomposition

High-dimensional token embeddings underpin Large Language Models (LLMs), as they can capture subtle semantic information and significantly enhance the modelling of complex language patterns. However, the associated high dimensionality also introduces considerable model parameters, and a prohibitively high model storage. To address this issue, this work proposes an approach based on the Tensor-Train Decomposition (TTD), where each token embedding is treated as a Matrix Product State (MPS) that can be efficiently computed in a distributed manner. The experimental results on GPT-2 demonstrate that, through our approach, the embedding layer can be compressed by a factor of up to 38.40 times, and when the compression factor is 3.31 times, even produced a better performance than the original GPT-2 model.

A Deep Matched Filter For R-Peak Detection in Ear-ECG

The Ear-ECG provides a continuous Lead I electrocardiogram (ECG) by measuring the potential difference related to heart activity using electrodes that can be embedded within earphones. The significant increase in wearability and comfort afforded by Ear-ECG is often accompanied by a corresponding degradation in signal quality - a common obstacle that is shared by most wearable technologies. We aim to resolve this issue by introducing a Deep Matched Filter (Deep-MF) for the highly accurate detection of R-peaks in wearable ECG, thus enhancing the utility of Ear-ECG in real-world scenarios. The Deep-MF consists of an encoder stage (trained as part of an encoder-decoder module to reproduce ground truth ECG), and an R-peak classifier stage. Through its operation as a Matched Filter, the encoder searches for matches with an ECG template pattern in the input signal, prior to filtering the matches with the subsequent convolutional layers and selecting peaks corresponding to true ECG matches. The so condensed latent representation of R-peak information is then fed into a simple R-peak classifier, of which the output provides precise R-peak locations. The proposed Deep Matched Filter is evaluated using leave-one-subject-out cross validation over 36 subjects with an age range of 18-75, with the Deep-MF outperforming existing algorithms for R-peak detection in noisy ECG. The Deep-MF achieves a median R-peak recall of 94.9\%, a median precision of 91.2\% and an (AUC) value of 0.97. Furthermore, we demonstrate that the Deep Matched Filter algorithm not only retains the initialised ECG kernel structure during the training process, but also amplifies portions of the ECG which it deems most valuable. Overall, the Deep Matched Filter serves as a valuable step forward for the real-world functionality of Ear-ECG and, through its explainable operation, the acceptance of deep learning models in e-health.

Amplitude-Independent Machine Learning for PPG through Visibility Graphs and Transfer Learning

Photoplethysmography (PPG) signals are omnipresent in wearable devices, as they measure blood volume variations using LED technology. These signals provide insight into the body's circulatory system and can be employed to extract various bio-features, such as heart rate and vascular ageing. Although several algorithms have been proposed for this purpose, many exhibit limitations, including heavy reliance on human calibration, high signal quality requirements, and a lack of generalization. In this paper, we introduce a PPG signal processing framework that integrates graph theory and computer vision algorithms, which is invariant to affine transformations, offers rapid computation speed, and exhibits robust generalization across tasks and datasets.