Determining the memory capacity of two-layer neural networks with m hidden neurons and input dimension d (i.e., md+m total trainable parameters), which refers to the largest size of general data the network can memorize, is a fundamental machine-learning question. For non-polynomial real analytic activation functions, such as sigmoids and smoothed rectified linear units (smoothed ReLUs), we establish a lower bound of md/2 and optimality up to a factor of approximately 2. Analogous prior results were limited to Heaviside and ReLU activations, with results for smooth activations suffering from logarithmic factors and requiring random data. To analyze the memory capacity, we examine the rank of the network's Jacobian by computing the rank of matrices involving both Hadamard powers and the Khati-Rao product. Our computation extends classical linear algebraic facts about the rank of Hadamard powers. Overall, our approach differs from previous works on memory capacity and holds promise for extending to deeper models and other architectures.
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Validation metrics are key for the reliable tracking of scientific progress and for bridging the current chasm between artificial intelligence (AI) research and its translation into practice. However, increasing evidence shows that particularly in image analysis, metrics are often chosen inadequately in relation to the underlying research problem. This could be attributed to a lack of accessibility of metric-related knowledge: While taking into account the individual strengths, weaknesses, and limitations of validation metrics is a critical prerequisite to making educated choices, the relevant knowledge is currently scattered and poorly accessible to individual researchers. Based on a multi-stage Delphi process conducted by a multidisciplinary expert consortium as well as extensive community feedback, the present work provides the first reliable and comprehensive common point of access to information on pitfalls related to validation metrics in image analysis. Focusing on biomedical image analysis but with the potential of transfer to other fields, the addressed pitfalls generalize across application domains and are categorized according to a newly created, domain-agnostic taxonomy. To facilitate comprehension, illustrations and specific examples accompany each pitfall. As a structured body of information accessible to researchers of all levels of expertise, this work enhances global comprehension of a key topic in image analysis validation.
A problem related to the development of algorithms designed to find the structure of artificial neural network used for behavioural (black-box) modelling of selected dynamic processes has been addressed in this paper. The research has included four original proposals of algorithms dedicated to neural network architecture search. Algorithms have been based on well-known optimisation techniques such as evolutionary algorithms and gradient descent methods. In the presented research an artificial neural network of recurrent type has been used, whose architecture has been selected in an optimised way based on the above-mentioned algorithms. The optimality has been understood as achieving a trade-off between the size of the neural network and its accuracy in capturing the response of the mathematical model under which it has been learnt. During the optimisation, original specialised evolutionary operators have been proposed. The research involved an extended validation study based on data generated from a mathematical model of the fast processes occurring in a pressurised water nuclear reactor.
Our research proposes a novel method for reducing the dimensionality of functional data, specifically for the case where the response is a scalar and the predictor is a random function. Our method utilizes distance covariance, and has several advantages over existing methods. Unlike current techniques which require restrictive assumptions such as linear conditional mean and constant covariance, our method has mild requirements on the predictor. Additionally, our method does not involve the use of the unbounded inverse of the covariance operator. The link function between the response and predictor can be arbitrary, and our proposed method maintains the advantage of being model-free, without the need to estimate the link function. Furthermore, our method is naturally suited for sparse longitudinal data. We utilize functional principal component analysis with truncation as a regularization mechanism in the development of our method. We provide justification for the validity of our proposed method, and establish statistical consistency of the estimator under certain regularization conditions. To demonstrate the effectiveness of our proposed method, we conduct simulation studies and real data analysis. The results show improved performance compared to existing methods.
We investigate the combinatorics of max-pooling layers, which are functions that downsample input arrays by taking the maximum over shifted windows of input coordinates, and which are commonly used in convolutional neural networks. We obtain results on the number of linearity regions of these functions by equivalently counting the number of vertices of certain Minkowski sums of simplices. We characterize the faces of such polytopes and obtain generating functions and closed formulas for the number of vertices and facets in a 1D max-pooling layer depending on the size of the pooling windows and stride, and for the number of vertices in a special case of 2D max-pooling.
Weak-label learning is a challenging task that requires learning from data "bags" containing positive and negative instances, but only the bag labels are known. The pool of negative instances is usually larger than positive instances, thus making selecting the most informative negative instance critical for performance. Such a selection strategy for negative instances from each bag is an open problem that has not been well studied for weak-label learning. In this paper, we study several sampling strategies that can measure the usefulness of negative instances for weak-label learning and select them accordingly. We test our method on CIFAR-10 and AudioSet datasets and show that it improves the weak-label classification performance and reduces the computational cost compared to random sampling methods. Our work reveals that negative instances are not all equally irrelevant, and selecting them wisely can benefit weak-label learning.
Including information from additional spectral bands (e.g., near-infrared) can improve deep learning model performance for many vision-oriented tasks. There are many possible ways to incorporate this additional information into a deep learning model, but the optimal fusion strategy has not yet been determined and can vary between applications. At one extreme, known as "early fusion," additional bands are stacked as extra channels to obtain an input image with more than three channels. At the other extreme, known as "late fusion," RGB and non-RGB bands are passed through separate branches of a deep learning model and merged immediately before a final classification or segmentation layer. In this work, we characterize the performance of a suite of multispectral deep learning models with different fusion approaches, quantify their relative reliance on different input bands and evaluate their robustness to naturalistic image corruptions affecting one or more input channels.
A rigidity circuit (in 2D) is a minimal dependent set in the rigidity matroid, i.e. a minimal graph supporting a non-trivial stress in any generic placement of its vertices in $\mathbb R^2$. Any rigidity circuit on $n\geq 5$ vertices can be obtained from rigidity circuits on a fewer number of vertices by applying the combinatorial resultant (CR) operation. The inverse operation is called a combinatorial resultant decomposition (CR-decomp). Any rigidity circuit on $n\geq 5$ vertices can be successively decomposed into smaller circuits, until the complete graphs $K_4$ are reached. This sequence of CR-decomps has the structure of a rooted binary tree called the combinatorial resultant tree (CR-tree). A CR-tree encodes an elimination strategy for computing circuit polynomials via Sylvester resultants. Different CR-trees lead to elimination strategies that can vary greatly in time and memory consumption. It is an open problem to establish criteria for optimal CR-trees, or at least to characterize those CR-trees that lead to good elimination strategies. In [12] we presented an algorithm for enumerating CR-trees where we give the algorithms for decomposing 3-connected rigidity circuits in polynomial time. In this paper we focus on those circuits that are not 3-connected, which we simply call 2-connected. In order to enumerate CR-decomps of 2-connected circuits $G$, a brute force exp-time search has to be performed among the subgraphs induced by the subsets of $V(G)$. This exp-time bottleneck is not present in the 3-connected case. In this paper we will argue that we do not have to account for all possible CR-decomps of 2-connected rigidity circuits to find a good elimination strategy; we only have to account for those CR-decomps that are a 2-split, all of which can be enumerated in polynomial time. We present algorithms and computational evidence in support of this heuristic.
Neuromorphic computing is one of the few current approaches that have the potential to significantly reduce power consumption in Machine Learning and Artificial Intelligence. Imam & Cleland presented an odour-learning algorithm that runs on a neuromorphic architecture and is inspired by circuits described in the mammalian olfactory bulb. They assess the algorithm's performance in "rapid online learning and identification" of gaseous odorants and odorless gases (short "gases") using a set of gas sensor recordings of different odour presentations and corrupting them by impulse noise. We replicated parts of the study and discovered limitations that affect some of the conclusions drawn. First, the dataset used suffers from sensor drift and a non-randomised measurement protocol, rendering it of limited use for odour identification benchmarks. Second, we found that the model is restricted in its ability to generalise over repeated presentations of the same gas. We demonstrate that the task the study refers to can be solved with a simple hash table approach, matching or exceeding the reported results in accuracy and runtime. Therefore, a validation of the model that goes beyond restoring a learned data sample remains to be shown, in particular its suitability to odour identification tasks.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.
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