We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.

The interpolation of spatial data can be of tremendous value in various applications, such as forecasting weather from only a few measurements of meteorological or remote sensing data. Existing methods for spatial interpolation, such as variants of kriging and spatial autoregressive models, tend to suffer from at least one of the following limitations: (a) the assumption of stationarity, (b) the assumption of isotropy, and (c) the trade-off between modelling local or global spatial interaction. Addressing these issues in this work, we propose the use of Markov reward processes (MRPs) as a spatial interpolation method, and we introduce three variants thereof: (i) a basic static discount MRP (SD-MRP), (ii) an accurate but mostly theoretical optimised MRP (O-MRP), and (iii) a transferable weight prediction MRP (WP-MRP). All variants of MRP interpolation operate locally, while also implicitly accounting for global spatial relationships in the entire system through recursion. Additionally, O-MRP and WP-MRP no longer assume stationarity and are robust to anisotropy. We evaluated our proposed methods by comparing the mean absolute errors of their interpolated grid cells to those of 7 common baselines, selected from models based on spatial autocorrelation, (spatial) regression, and deep learning. We performed detailed evaluations on two publicly available datasets (local GDP values, and COVID-19 patient trajectory data). The results from these experiments clearly show the competitive advantage of MRP interpolation, which achieved significantly lower errors than the existing methods in 23 out of 40 experimental conditions, or 35 out of 40 when including O-MRP.

Imitation learning enables high-fidelity, vision-based learning of policies within rich, photorealistic environments. However, such techniques often rely on traditional discrete-time neural models and face difficulties in generalizing to domain shifts by failing to account for the causal relationships between the agent and the environment. In this paper, we propose a theoretical and experimental framework for learning causal representations using continuous-time neural networks, specifically over their discrete-time counterparts. We evaluate our method in the context of visual-control learning of drones over a series of complex tasks, ranging from short- and long-term navigation, to chasing static and dynamic objects through photorealistic environments. Our results demonstrate that causal continuous-time deep models can perform robust navigation tasks, where advanced recurrent models fail. These models learn complex causal control representations directly from raw visual inputs and scale to solve a variety of tasks using imitation learning.

Handwritten digit recognition is one of the extensively studied area in machine learning. Apart from the wider research on handwritten digit recognition on MNIST dataset, there are many other research works on various script recognition. However, it is not very common for multi-script digit recognition which encourage the development of robust and multipurpose systems. Additionally working on multi-script digit recognition enables multi-task learning, considering the script classification as a related task for instance. It is evident that multi-task learning improves model performance through inductive transfer using the information contained in related tasks. Therefore, in this study multi-script handwritten digit recognition using multi-task learning will be investigated. As a specific case of demonstrating the solution to the problem, Amharic handwritten character recognition will also be experimented. The handwritten digits of three scripts including Latin, Arabic and Kannada are studied to show that multi-task models with reformulation of the individual tasks have shown promising results. In this study a novel way of using the individual tasks predictions was proposed to help classification performance and regularize the different loss for the purpose of the main task. This finding has outperformed the baseline and the conventional multi-task learning models. More importantly, it avoided the need for weighting the different losses of the tasks, which is one of the challenges in multi-task learning.

The activations of language transformers like GPT-2 have been shown to linearly map onto brain activity during speech comprehension. However, the nature of these activations remains largely unknown and presumably conflate distinct linguistic classes. Here, we propose a taxonomy to factorize the high-dimensional activations of language models into four combinatorial classes: lexical, compositional, syntactic, and semantic representations. We then introduce a statistical method to decompose, through the lens of GPT-2's activations, the brain activity of 345 subjects recorded with functional magnetic resonance imaging (fMRI) during the listening of ~4.6 hours of narrated text. The results highlight two findings. First, compositional representations recruit a more widespread cortical network than lexical ones, and encompass the bilateral temporal, parietal and prefrontal cortices. Second, contrary to previous claims, syntax and semantics are not associated with separated modules, but, instead, appear to share a common and distributed neural substrate. Overall, this study introduces a versatile framework to isolate, in the brain activity, the distributed representations of linguistic constructs.

Computational Colour Constancy (CCC) consists of estimating the colour of one or more illuminants in a scene and using them to remove unwanted chromatic distortions. Much research has focused on illuminant estimation for CCC on single images, with few attempts of leveraging the temporal information intrinsic in sequences of correlated images (e.g., the frames in a video), a task known as Temporal Colour Constancy (TCC). The state-of-the-art for TCC is TCCNet, a deep-learning architecture that uses a ConvLSTM for aggregating the encodings produced by CNN submodules for each image in a sequence. We extend this architecture with different models obtained by (i) substituting the TCCNet submodules with C4, the state-of-the-art method for CCC targeting images; (ii) adding a cascading strategy to perform an iterative improvement of the estimate of the illuminant. We tested our models on the recently released TCC benchmark and achieved results that surpass the state-of-the-art. Analyzing the impact of the number of frames involved in illuminant estimation on performance, we show that it is possible to reduce inference time by training the models on few selected frames from the sequences while retaining comparable accuracy.

Learning to interact with the environment not only empowers the agent with manipulation capability but also generates information to facilitate building of action understanding and imitation capabilities. This seems to be a strategy adopted by biological systems, in particular primates, as evidenced by the existence of mirror neurons that seem to be involved in multi-modal action understanding. How to benefit from the interaction experience of the robots to enable understanding actions and goals of other agents is still a challenging question. In this study, we propose a novel method, deep modality blending networks (DMBN), that creates a common latent space from multi-modal experience of a robot by blending multi-modal signals with a stochastic weighting mechanism. We show for the first time that deep learning, when combined with a novel modality blending scheme, can facilitate action recognition and produce structures to sustain anatomical and effect-based imitation capabilities. Our proposed system, can be conditioned on any desired sensory/motor value at any time-step, and can generate a complete multi-modal trajectory consistent with the desired conditioning in parallel avoiding accumulation of prediction errors. We further showed that given desired images from different perspectives, i.e. images generated by the observation of other robots placed on different sides of the table, our system could generate image and joint angle sequences that correspond to either anatomical or effect based imitation behavior. Overall, the proposed DMBN architecture not only serves as a computational model for sustaining mirror neuron-like capabilities, but also stands as a powerful machine learning architecture for high-dimensional multi-modal temporal data with robust retrieval capabilities operating with partial information in one or multiple modalities.

This paper presents a new data-driven framework for analyzing periodic physical human-robot interaction (pHRI) in latent state space. To elaborate human understanding and/or robot control during pHRI, the model representing pHRI is critical. Recent developments of deep learning technologies would enable us to learn such a model from a dataset collected from the actual pHRI. Our framework is developed based on variational recurrent neural network (VRNN), which can inherently handle time-series data like one pHRI generates. This paper modifies VRNN in order to include the latent dynamics from robot to human explicitly. In addition, to analyze periodic motions like walking, we integrate a new recurrent network based on reservoir computing (RC), which has random and fixed connections between numerous neurons, with VRNN. By augmenting RC into complex domain, periodic behavior can be represented as the phase rotation in complex domain without decaying the amplitude. For verification of the proposed framework, a rope-rotation/swinging experiment was analyzed. The proposed framework, trained on the dataset collected from the experiment, achieved the latent state space where the differences in periodic motions can be distinguished. Such a well-distinguished space yielded the best prediction accuracy of the human observations and the robot actions. The attached video can be seen in youtube: https://youtu.be/umn0MVcIpsY

Recently, linear regression models, such as EASE and SLIM, have shown to often produce rather competitive results against more sophisticated deep learning models. On the other side, the (weighted) matrix factorization approaches have been popular choices for recommendation in the past and widely adopted in the industry. In this work, we aim to theoretically understand the relationship between these two approaches, which are the cornerstones of model-based recommendations. Through the derivation and analysis of the closed-form solutions for two basic regression and matrix factorization approaches, we found these two approaches are indeed inherently related but also diverge in how they "scale-down" the singular values of the original user-item interaction matrix. This analysis also helps resolve the questions related to the regularization parameter range and model complexities. We further introduce a new learning algorithm in searching (hyper)parameters for the closed-form solution and utilize it to discover the nearby models of the existing solutions. The experimental results demonstrate that the basic models and their closed-form solutions are indeed quite competitive against the state-of-the-art models, thus, confirming the validity of studying the basic models. The effectiveness of exploring the nearby models are also experimentally validated.

Explainable AI (XAI) methods are used in order to improve the interpretability of these complex "black box" models, thereby increasing transparency and enabling informed decision-making (Guidotti et al, 2018). Despite this, methods to assess the quality of explanations generated by such explainable methods are so far under-explored. In particular, functionallygrounded evaluation methods, which measure the inherent ability of explainable methods in a given situation, are often specific to a particular type of dataset or explainable method. A key measure of functionally-grounded explanation fitness is explanation fidelity, which assesses the correctness and completeness of the explanation with respect to the underlying black box predictive model (Zhou et al, 2021). Evaluations of fidelity in literature can generally be classified as one of the following: external fidelity evaluation, which assesses how well the prediction of the underlying model and the explanation agree, and internal fidelity, which assesses how well the explanation matches the decision-making processes of the underlying model (Messalas et al, 2019). While methods to evaluate external fidelity are relatively common in literature (Guidotti et al, 2019; Lakkaraju et al, 2016; Ming et al, 2019; Shankaranarayana and Runje, 2019), evaluation methods to evaluate internal fidelity using black box models are generally limited to text and image data, rather than tabular (Du et al, 2019; Fong and Vedaldi, 2017; Nguyen, 2018; Samek et al, 2017). In this paper, weproposeanovelevaluation method based onathree phase approach:(1) the creation of a fully transparent, inherently interpretable white box model, and evaluation of explanations against this model; (2) the usage of the white box as a proxy to refine and improve the evaluation of explanations generated by a black box model; and (3) test the fidelity of explanations for a black box model using the refined method from the second phase. The main contributions of this work are as follows: 1.