We thank the reviewers for their feedback. We will first respond to shared and then to individual comments. Additionally, reviewers 2 and 3 requested clarification regarding the advantages of DCA over other methods. For instance, one could attempt to correlate each neuron's contribution to the DCA subspace with single-neuron Studying the behavior of Kernel DCA is a direction for future studies. Additionally, we found and corrected a minor bug in Figure 1A: the SFA and DCA lines are now blue and red, respectively.

Based on this bound, our detector continuously monitors the operation of the network over a test window and fires off an alarm whenever a deviation is detected.

We thank the reviewers for their insightful feedback! Our goal is not to reduce the dimensionality further below n . What are the convergence properties of the proposed training method (R4)? Is the sequential or alternating training scheme better (R4)? It would be nice to have a different metric to compare the models (R1).

Summarizing high-dimensional data using a small number of parameters is a ubiquitous first step in the analysis of neuronal population activity. Recently developed methods use targeted approaches that work by identifying multiple, distinct low-dimensional subspaces of activity that capture the population response to individual experimental task variables, such as the value of a presented stimulus or the behavior of the animal. These methods have gained attention because they decompose total neural activity into what are ostensibly different parts of a neuronal computation. However, existing targeted methods have been developed outside of the confines of probabilistic modeling, making some aspects of the procedures ad hoc, or limited in flexibility or interpretability. Here we propose a new model-based method for targeted dimensionality reduction based on a probabilistic generative model of the population response data.

Dimensionality reduction plays a central role in real-world applications for Machine Learning, among many fields. In particular, metric dimensionality reduction where data from a general metric is mapped into low dimensional space, is often used as a first step before applying machine learning algorithms. In almost all these applications the quality of the embedding is measured by various average case criteria. Metric dimensionality reduction has also been studied in Math and TCS, within the extremely fruitful and influential field of metric embedding. Yet, the vast majority of theoretical research has been devoted to analyzing the worst case behavior of embeddings and therefore has little relevance to practical settings.

We developed a Nonlinear Level-set Learning (NLL) method for dimensionality reduction in high-dimensional function approximation with small data. This work is motivated by a variety of design tasks in real-world engineering applications, where practitioners would replace their computationally intensive physical models (e.g., high-resolution fluid simulators) with fast-to-evaluate predictive machine learning models, so as to accelerate the engineering design processes. There are two major challenges in constructing such predictive models: (a) high-dimensional inputs (e.g., many independent design parameters) and (b) small training data, generated by running extremely time-consuming simulations. Thus, reducing the input dimension is critical to alleviate the over-fitting issue caused by data insufficiency. Existing methods, including sliced inverse regression and active subspace approaches, reduce the input dimension by learning a linear coordinate transformation; our main contribution is to extend the transformation approach to a nonlinear regime. Specifically, we exploit reversible networks (RevNets) to learn nonlinear level sets of a high-dimensional function and parameterize its level sets in low-dimensional spaces. A new loss function was designed to utilize samples of the target functions' gradient to encourage the transformed function to be sensitive to only a few transformed coordinates. The NLL approach is demonstrated by applying it to three 2D functions and two 20D functions for showing the improved approximation accuracy with the use of nonlinear transformation, as well as to an 8D composite material design problem for optimizing the buckling-resistance performance of composite shells of rocket inter-stages.

The Wasserstein barycenter is a geometric construct which captures the notion of centrality among probability distributions, and which has found many applications in machine learning. However, most algorithms for finding even an approximate barycenter suffer an exponential dependence on the dimension $d$ of the underlying space of the distributions. In order to cope with this ``curse of dimensionality,'' we study dimensionality reduction techniques for the Wasserstein barycenter problem. When the barycenter is restricted to support of size $n$, we show that randomized dimensionality reduction can be used to map the problem to a space of dimension $O(\log n)$ independent of both $d$ and $k$, and that \emph{any} solution found in the reduced dimension will have its cost preserved up to arbitrary small error in the original space. We provide matching upper and lower bounds on the size of the reduced dimension, showing that our methods are optimal up to constant factors. We also provide a coreset construction for the Wasserstein barycenter problem that significantly decreases the number of input distributions. The coresets can be used in conjunction with random projections and thus further improve computation time. Lastly, our experimental results validate the speedup provided by dimensionality reduction while maintaining solution quality.

Traditional sea exploration faces significant challenges due to extreme conditions, limited visibility, and high costs, resulting in vast unexplored ocean regions. This paper presents an innovative AI-powered Autonomous Underwater Vehicle (AUV) system designed to overcome these limitations by automating underwater object detection, analysis, and reporting. The system integrates YOLOv12 Nano for real-time object detection, a Convolutional Neural Network (CNN) (ResNet50) for feature extraction, Principal Component Analysis (PCA) for dimensionality reduction, and K-Means++ clustering for grouping marine objects based on visual characteristics. Furthermore, a Large Language Model (LLM) (GPT-4o Mini) is employed to generate structured reports and summaries of underwater findings, enhancing data interpretation. The system was trained and evaluated on a combined dataset of over 55,000 images from the DeepFish and OzFish datasets, capturing diverse Australian marine environments. Experimental results demonstrate the system's capability to detect marine objects with a mAP@0.5 of 0.512, a precision of 0.535, and a recall of 0.438. The integration of PCA effectively reduced feature dimensionality while preserving 98% variance, facilitating K-Means clustering which successfully grouped detected objects based on visual similarities. The LLM integration proved effective in generating insightful summaries of detections and clusters, supported by location data. This integrated approach significantly reduces the risks associated with human diving, increases mission efficiency, and enhances the speed and depth of underwater data analysis, paving the way for more effective scientific research and discovery in challenging marine environments.

The Information Contrastive (I-Con) framework revealed that over 23 representation learning methods implicitly minimize KL divergence between data and learned distributions that encode similarities between data points. However, a KL-based loss may be misaligned with the true objective, and properties of KL divergence such as asymmetry and unboundedness may create optimization challenges. We present Beyond I-Con, a framework that enables systematic discovery of novel loss functions by exploring alternative statistical divergences. Key findings: (1) on unsupervised clustering of DINO-ViT embeddings, we achieve state-of-the-art results by modifying the PMI algorithm to use total variation (TV) distance; (2) supervised contrastive learning with Euclidean distance as the feature space metric is improved by replacing the standard loss function with Jenson-Shannon divergence (JSD); (3) on dimensionality reduction, we achieve superior qualitative results and better performance on downstream tasks than SNE by replacing KL with a bounded $f$-divergence. Our results highlight the importance of considering divergence choices in representation learning optimization.

This paradigm challenges traditional data management and analysis techniques by demanding innovative solutions capable of processing, analyzing, and deriving insights from vast and diverse datasets. In particular, the inclusion of mixed data types, such as numerical and categorical variables, poses significant challenges to conventional methodologies, necessitating the development of novel approaches to effectively leverage the wealth of information available [2]. Traditionally, data handling methods were designed around homogeneous datasets, typically consisting of numerical values. However, the big data paradigm introduces a multitude of data types, including structured, unstructured, and semi-structured data, which demand a departure from traditional approaches. Moreover, the three primary characteristics of big data--volume, velocity, and variety--amplify the complexity of data analysis, requiring scalable and adaptable solutions capable of processing large volumes of data at high speeds while accommodating diverse data formats and structures. These methods for handling mixed data often involve separate analyses of categorical and numerical variables, treating them as distinct entities rather than integrating their interdependencies.