Low-Rank Learning by Design: the Role of Network Architecture and Activation Linearity in Gradient Rank Collapse

Our understanding of learning dynamics of deep neural networks (DNNs) remains incomplete. Recent research has begun to uncover the mathematical principles underlying these networks, including the phenomenon of "Neural Collapse", where linear classifiers within DNNs converge to specific geometrical structures during late-stage training. However, the role of geometric constraints in learning extends beyond this terminal phase. For instance, gradients in fully-connected layers naturally develop a low-rank structure due to the accumulation of rank-one outer products over a training batch. Despite the attention given to methods that exploit this structure for memory saving or regularization, the emergence of low-rank learning as an inherent aspect of certain DNN architectures has been under-explored. In this paper, we conduct a comprehensive study of gradient rank in DNNs, examining how architectural choices and structure of the data effect gradient rank bounds. Our theoretical analysis provides these bounds for training fully-connected, recurrent, and convolutional neural networks. We also demonstrate, both theoretically and empirically, how design choices like activation function linearity, bottleneck layer introduction, convolutional stride, and sequence truncation influence these bounds. Our findings not only contribute to the understanding of learning dynamics in DNNs, but also provide practical guidance for deep learning engineers to make informed design decisions.

Comparing Differentiable Logics for Learning Systems: A Research Preview

Extensive research on formal verification of machine learning (ML) systems indicates that learning from data alone often fails to capture underlying background knowledge. A variety of verifiers have been developed to ensure that a machine-learnt model satisfies correctness and safety properties, however, these verifiers typically assume a trained network with fixed weights. ML-enabled autonomous systems are required to not only detect incorrect predictions, but should also possess the ability to self-correct, continuously improving and adapting. A promising approach for creating ML models that inherently satisfy constraints is to encode background knowledge as logical constraints that guide the learning process via so-called differentiable logics. In this research preview, we compare and evaluate various logics from the literature in weakly-supervised contexts, presenting our findings and highlighting open problems for future work. Our experimental results are broadly consistent with results reported previously in literature; however, learning with differentiable logics introduces a new hyperparameter that is difficult to tune and has significant influence on the effectiveness of the logics.

Gradients without Backpropagation

Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic differentiation algorithms that also includes the forward mode. We present a method to compute gradients based solely on the directional derivative that one can compute exactly and efficiently via the forward mode. We call this formulation the forward gradient, an unbiased estimate of the gradient that can be evaluated in a single forward run of the function, entirely eliminating the need for backpropagation in gradient descent. We demonstrate forward gradient descent in a range of problems, showing substantial savings in computation and enabling training up to twice as fast in some cases.

Continuous Convolutional Neural Networks: Coupled Neural PDE and ODE

Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden dynamics of a physical system using ordinary differential equation (ODEs) systems (ODEs) and Partial Differential Equation systems (PDEs). Instead of considering the physical system such as image, time -series as a system of multiple layers, this new technique can model a system in the form of Differential Equation (DEs). The proposed method has been assessed by solving several steady-state PDEs on irregular domains, including heat equations, Navier-Stokes equations.

Neural Network based on Automatic Differentiation Transformation of Numeric Iterate-to-Fixedpoint

This work proposes a Neural Network model that can control its depth using an iterate-to-fixed-point operator. The architecture starts with a standard layered Network but with added connections from current later to earlier layers, along with a gate to make them inactive under most circumstances. These ``temporal wormhole'' connections create a shortcut that allows the Neural Network to use the information available at deeper layers and re-do earlier computations with modulated inputs. End-to-end training is accomplished by using appropriate calculations for a numeric iterate-to-fixed-point operator. In a typical case, where the ``wormhole'' connections are inactive, this is inexpensive; but when they are active, the network takes a longer time to settle down, and the gradient calculation is also more laborious, with an effect similar to making the network deeper. In contrast to the existing skip-connection concept, this proposed technique enables information to flow up and down in the network. Furthermore, the flow of information follows a fashion that seems analogous to the afferent and efferent flow of information through layers of processing in the brain. We evaluate models that use this novel mechanism on different long-term dependency tasks. The results are competitive with other studies, showing that the proposed model contributes significantly to overcoming traditional deep learning models' vanishing gradient descent problem. At the same time, the training time is significantly reduced, as the ``easy'' input cases are processed more quickly than ``difficult'' ones.

ECG synthesis with Neural ODE and GAN models

Continuous medical time series data such as ECG is one of the most complex time series due to its dynamic and high dimensional characteristics. In addition, due to its sensitive nature, privacy concerns and legal restrictions, it is often even complex to use actual data for different medical research. As a result, generating continuous medical time series is a very critical research area. Several research works already showed that the ability of generative adversarial networks (GANs) in the case of continuous medical time series generation is promising. Most medical data generation works, such as ECG synthesis, are mainly driven by the GAN model and its variation. On the other hand, Some recent work on Neural Ordinary Differential Equation (Neural ODE) demonstrates its strength against informative missingness, high dimension as well as dynamic nature of continuous time series. Instead of considering continuous-time series as a discrete-time sequence, Neural ODE can train continuous time series in real-time continuously. In this work, we used Neural ODE based model to generate synthetic sine waves and synthetic ECG. We introduced a new technique to design the generative adversarial network with Neural ODE based Generator and Discriminator. We developed three new models to synthesise continuous medical data. Different evaluation metrics are then used to quantitatively assess the quality of generated synthetic data for real-world applications and data analysis. Another goal of this work is to combine the strength of GAN and Neural ODE to generate synthetic continuous medical time series data such as ECG. We also evaluated both the GAN model and the Neural ODE model to understand the comparative efficiency of models from the GAN and Neural ODE family in medical data synthesis.

HeunNet: Extending ResNet using Heun's Methods

There is an analogy between the ResNet (Residual Network) architecture for deep neural networks and an Euler solver for an ODE. The transformation performed by each layer resembles an Euler step in solving an ODE. We consider the Heun Method, which involves a single predictor-corrector cycle, and complete the analogy, building a predictor-corrector variant of ResNet, which we call a HeunNet. Just as Heun's method is more accurate than Euler's, experiments show that HeunNet achieves high accuracy with low computational (both training and test) time compared to both vanilla recurrent neural networks and other ResNet variants.

Neural Ordinary Differential Equation based Recurrent Neural Network Model

Neural differential equations are a promising new member in the neural network family. They show the potential of differential equations for time series data analysis. In this paper, the strength of the ordinary differential equation (ODE) is explored with a new extension. The main goal of this work is to answer the following questions: (i)~can ODE be used to redefine the existing neural network model? (ii)~can Neural ODEs solve the irregular sampling rate challenge of existing neural network models for a continuous time series, i.e., length and dynamic nature, (iii)~how to reduce the training and evaluation time of existing Neural ODE systems? This work leverages the mathematical foundation of ODEs to redesign traditional RNNs such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU). The main contribution of this paper is to illustrate the design of two new ODE-based RNN models (GRU-ODE model and LSTM-ODE) which can compute the hidden state and cell state at any point of time using an ODE solver. These models reduce the computation overhead of hidden state and cell state by a vast amount. The performance evaluation of these two new models for learning continuous time series with irregular sampling rate is then demonstrated. Experiments show that these new ODE based RNN models require less training time than Latent ODEs and conventional Neural ODEs. They can achieve higher accuracy quickly, and the design of the neural network is simpler than, previous neural ODE systems.

Neural ODEs for Informative Missingness in Multivariate Time Series

Informative missingness is unavoidable in the digital processing of continuous time series, where the value for one or more observations at different time points are missing. Such missing observations are one of the major limitations of time series processing using deep learning. Practical applications, e.g., sensor data, healthcare, weather, generates data that is in truth continuous in time, and informative missingness is a common phenomenon in these datasets. These datasets often consist of multiple variables, and often there are missing values for one or many of these variables. This characteristic makes time series prediction more challenging, and the impact of missing input observations on the accuracy of the final output can be significant. A recent novel deep learning model called GRU-D is one early attempt to address informative missingness in time series data. On the other hand, a new family of neural networks called Neural ODEs (Ordinary Differential Equations) are natural and efficient for processing time series data which is continuous in time. In this paper, a deep learning model is proposed that leverages the effective imputation of GRU-D, and the temporal continuity of Neural ODEs. A time series classification task performed on the PhysioNet dataset demonstrates the performance of this architecture.