When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called "neural collapse" phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer's classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural collapse phenomenon for normalized features. Based on an unconstrained feature model, we simplify the empirical loss function in a multi-class classification task into a nonconvex optimization problem over the Riemannian manifold by constraining all features and classifiers over the sphere. In this context, we analyze the nonconvex landscape of the Riemannian optimization problem over the product of spheres, showing a benign global landscape in the sense that the only global minimizers are the neural collapse solutions while all other critical points are strict saddles with negative curvature. Experimental results on practical deep networks corroborate our theory and demonstrate that better representations can be learned faster via feature normalization.

We study optimal transport-based distributionally robust optimization problems where a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by reshaping a prescribed reference distribution at a finite transportation cost. In this framework, we show that robustification is intimately related to various forms of variation and Lipschitz regularization even if the transportation cost function fails to be (some power of) a metric. We also derive conditions for the existence and the computability of a Nash equilibrium between the decision-maker and nature, and we demonstrate numerically that nature's Nash strategy can be viewed as a distribution that is supported on remarkably deceptive adversarial samples. Finally, we identify practically relevant classes of optimal transport-based distributionally robust optimization problems that can be addressed with efficient gradient descent algorithms even if the loss function or the transportation cost function are nonconvex (but not both at the same time).

Future NASA lander missions to icy moons will require completely automated, accurate, and data efficient calibration methods for the robot manipulator arms that sample icy terrains in the lander's vicinity. To support this need, this paper presents a Gaussian Process (GP) approach to the classical manipulator kinematic calibration process. Instead of identifying a corrected set of Denavit-Hartenberg kinematic parameters, a set of GPs models the residual kinematic error of the arm over the workspace. More importantly, this modeling framework allows a Gaussian Process Upper Confident Bound (GP-UCB) algorithm to efficiently and adaptively select the calibration's measurement points so as to minimize the number of experiments, and therefore minimize the time needed for recalibration. The method is demonstrated in simulation on a simple 2-DOF arm, a 6 DOF arm whose geometry is a candidate for a future NASA mission, and a 7 DOF Barrett WAM arm.

Optimizing delivery routes for last-mile logistics service is challenging and has attracted the attention of many researchers. These problems are usually modeled and solved as variants of vehicle routing problems (VRPs) with challenging real-world constraints (e.g., time windows, precedence). However, despite many decades of solid research on solving these VRP instances, we still see significant gaps between optimized routes and the routes that are actually preferred by the practitioners. Most of these gaps are due to the difference between what's being optimized, and what the practitioners actually care about, which is hard to be defined exactly in many instances. In this paper, we propose a novel hierarchical route optimizer with learnable parameters that combines the strength of both the optimization and machine learning approaches. Our hierarchical router first solves a zone-level Traveling Salesman Problem with learnable weights on various zone-level features; with the zone visit sequence fixed, we then solve the stop-level vehicle routing problem as a Shortest Hamiltonian Path problem. The Bayesian optimization approach is then introduced to allow us to adjust the weights to be assigned to different zone features used in solving the zone-level Traveling Salesman Problem. By using a real-world delivery dataset provided by the Amazon Last Mile Routing Research Challenge, we demonstrate the importance of having both the optimization and the machine learning components. We also demonstrate how we can use route-related features to identify instances that we might have difficulty with. This paves ways to further research on how we can tackle these difficult instances.

The recently proposed Joint Energy-based Model (JEM) interprets discriminatively trained classifier $p(y|x)$ as an energy model, which is also trained as a generative model describing the distribution of the input observations $p(x)$. The JEM training relies on "positive examples" (i.e. examples from the training data set) as well as on "negative examples", which are samples from the modeled distribution $p(x)$ generated by means of Stochastic Gradient Langevin Dynamics (SGLD). Unfortunately, SGLD often fails to deliver negative samples of sufficient quality during the standard JEM training, which causes a very unbalanced contribution from the positive and negative examples when calculating gradients for JEM updates. As a consequence, the standard JEM training is quite unstable requiring careful tuning of hyper-parameters and frequent restarts when the training starts diverging. This makes it difficult to apply JEM to different neural network architectures, modalities, and tasks. In this work, we propose a training procedure that stabilizes SGLD-based JEM training (ST-JEM) by balancing the contribution from the positive and negative examples. We also propose to add an additional "regularization" term to the training objective -- MI between the input observations $x$ and output labels $y$ -- which encourages the JEM classifier to make more certain decisions about output labels. We demonstrate the effectiveness of our approach on the CIFAR10 and CIFAR100 tasks. We also consider the task of classifying phonemes in a speech signal, for which we were not able to train JEM without the proposed stabilization. We show that a convincing speech can be generated from the trained model. Alternatively, corrupted speech can be de-noised by bringing it closer to the modeled speech distribution using a few SGLD iterations. We also propose and discuss additional applications of the trained model.

A deep learning model, EscherNet 101, is constructed to categorize images of 2D periodic patterns into their respective 17 wallpaper groups. Beyond evaluating EscherNet 101 performance by classification rates, at a micro-level we investigate the filters learned at different layers in the network, capable of capturing second-order invariants beyond edge and curvature.

Decision trees are widely used for their low computational cost, good predictive performance, and ability to assess the importance of features. Though often used in practice for feature selection, the theoretical guarantees of these methods are not well understood. We here obtain a tight finite sample bound for the feature selection problem in linear regression using single-depth decision trees. We examine the statistical properties of these "decision stumps" for the recovery of the $s$ active features from $p$ total features, where $s \ll p$. Our analysis provides tight sample performance guarantees on high-dimensional sparse systems which align with the finite sample bound of $O(s \log p)$ as obtained by Lasso, improving upon previous bounds for both the median and optimal splitting criteria. Our results extend to the non-linear regime as well as arbitrary sub-Gaussian distributions, demonstrating that tree based methods attain strong feature selection properties under a wide variety of settings and further shedding light on the success of these methods in practice. As a byproduct of our analysis, we show that we can provably guarantee recovery even when the number of active features $s$ is unknown. We further validate our theoretical results and proof methodology using computational experiments.

It has been hypothesized that the ventral stream processing for object recognition is based on a mechanism called cortically local subspace untangling. A mathematical abstraction of object recognition by the visual cortex is how to untangle the manifolds associated with different object category. Such a manifold untangling problem is closely related to the celebrated kernel trick in metric space. In this paper, we conjecture that there is a more general solution to manifold untangling in the topological space without artificially defining any distance metric. Geometrically, we can either $embed$ a manifold in a higher dimensional space to promote selectivity or $flatten$ a manifold to promote tolerance. General strategies of both global manifold embedding and local manifold flattening are presented and connected with existing work on the untangling of image, audio, and language data. We also discuss the implications of untangling the manifold into motor control and internal representations.

Objective:Computer vision-based up-to-date accurate damage classification and localization are of decisive importance for infrastructure monitoring, safety, and the serviceability of civil infrastructure. Current state-of-the-art deep learning (DL)-based damage detection models, however, often lack superior feature extraction capability in complex and noisy environments, limiting the development of accurate and reliable object distinction. Method: To this end, we present DenseSPH-YOLOv5, a real-time DL-based high-performance damage detection model where DenseNet blocks have been integrated with the backbone to improve in preserving and reusing critical feature information. Additionally, convolutional block attention modules (CBAM) have been implemented to improve attention performance mechanisms for strong and discriminating deep spatial feature extraction that results in superior detection under various challenging environments. Moreover, additional feature fusion layers and a Swin-Transformer Prediction Head (SPH) have been added leveraging advanced self-attention mechanism for more efficient detection of multiscale object sizes and simultaneously reducing the computational complexity. Results: Evaluating the model performance in large-scale Road Damage Dataset (RDD-2018), at a detection rate of 62.4 FPS, DenseSPH-YOLOv5 obtains a mean average precision (mAP) value of 85.25 %, F1-score of 81.18 %, and precision (P) value of 89.51 % outperforming current state-of-the-art models. Significance: The present research provides an effective and efficient damage localization model addressing the shortcoming of existing DL-based damage detection models by providing highly accurate localized bounding box prediction. Current work constitutes a step towards an accurate and robust automated damage detection system in real-time in-field applications.

Despite the remarkable accuracies attained by machine learning classifiers to separate complex datasets in a supervised fashion, most of their operation falls short to provide an informed intuition about the structure of data, and, what is more important, about the phenomena being characterized by the given datasets. By contrast, topological data analysis (TDA) is devoted to study the shape of data clouds by means of persistence descriptors and provides a quantitative characterization of specific topological features of the dataset under scrutiny. In this article we introduce a novel TDA-based classifier that works on the principle of assessing quantifiable changes on topological metrics caused by the addition of new input to a subset of data. We used this classifier with a high-dimensional electro-encephalographic (EEG) dataset recorded from eleven participants during a decision-making experiment in which three motivational states were induced through a manipulation of social pressure. After processing a band-pass filtered version of EEG signals, we calculated silhouettes from persistence diagrams associated with each motivated state, and classified unlabeled signals according to their impact on each reference silhouette. Our results show that in addition to providing accuracies within the range of those of a nearest neighbour classifier, the TDA classifier provides formal intuition of the structure of the dataset as well as an estimate of its intrinsic dimension. Towards this end, we incorporated dimensionality reduction methods to our procedure and found that the accuracy of our TDA classifier is generally not sensitive to explained variance but rather to shape, contrary to what happens with most machine learning classifiers.