Instead, point clouds or continuous probability measures are the appropriate data structures. These data arise naturally in fields such as computer vision, image processing, shape analysis, and generative modeling, where representing complex objects as probability distributions provides a richer and more flexible framework for analysis. Real-world examples include text documents with bag-of-words models treating word counts as features, which forms a histogram for each document [35], imaging data where pixel intensity is interpreted as mass [26] and results in 2D discrete probability measures over the image grid, and gene expression data that is interpretted as a distribution across a gene network [8, 15]. Optimal transport (OT) theory [30] has recently emerged as a powerful tool to compare probability measures. Qualitatively, OT generates a distance metric between probability measures by minimizing the work needed to move one distribution into another over all transport plans. It has gained significant popularity for applications [4, 26, 27] involving point clouds and probability distributions. OT allows for the computation of distances between distributions by solving a minimization problem over transportation plans. Despite its theoretical elegance and its ability to capture geometric properties of distributions, using vanilla OT is computationally expensive and does not directly integrate into existing machine learning pipelines. For this reason, OT has been somewhat limited in practical applications, particularly in settings that demand scalable and efficient algorithms for tasks such as classification, dimension reduction, and generation.

We construct and analyze a neural network two-sample test to determine whether two datasets came from the same distribution (null hypothesis) or not (alternative hypothesis). We perform time-analysis on a neural tangent kernel (NTK) two-sample test. In particular, we derive the theoretical minimum training time needed to ensure the NTK two-sample test detects a deviation-level between the datasets. Similarly, we derive the theoretical maximum training time before the NTK two-sample test detects a deviation-level. By approximating the neural network dynamics with the NTK dynamics, we extend this time-analysis to the realistic neural network two-sample test generated from time-varying training dynamics and finite training samples. A similar extension is done for the neural network two-sample test generated from time-varying training dynamics but trained on the population. To give statistical guarantees, we show that the statistical power associated with the neural network two-sample test goes to 1 as the neural network training samples and test evaluation samples go to infinity. Additionally, we prove that the training times needed to detect the same deviation-level in the null and alternative hypothesis scenarios are well-separated. Finally, we run some experiments showcasing a two-layer neural network two-sample test on a hard two-sample test problem and plot a heatmap of the statistical power of the two-sample test in relation to training time and network complexity.

The last few years have witnessed great success on image generation, which has crossed the acceptance thresholds of aesthetics, making it directly applicable to personal and commercial applications. However, images, especially in marketing and advertising applications, are often created as a means to an end as opposed to just aesthetic concerns. The goal can be increasing sales, getting more clicks, likes, or image sales (in the case of stock businesses). Therefore, the generated images need to perform well on these key performance indicators (KPIs), in addition to being aesthetically good. In this paper, we make the first endeavor to answer the question of "How can one infuse the knowledge of the end-goal within the image generation process itself to create not just better-looking images but also "better-performing'' images?''. We propose BoigLLM, an LLM that understands both image content and user behavior. BoigLLM knows how an image should look to get a certain required KPI. We show that BoigLLM outperforms 13x larger models such as GPT-3.5 and GPT-4 in this task, demonstrating that while these state-of-the-art models can understand images, they lack information on how these images perform in the real world. To generate actual pixels of behavior-conditioned images, we train a diffusion-based model (BoigSD) to align with a proposed BoigLLM-defined reward. We show the performance of the overall pipeline on two datasets covering two different behaviors: a stock dataset with the number of forward actions as the KPI and a dataset containing tweets with the total likes as the KPI, denoted as BoigBench. To advance research in the direction of utility-driven image generation and understanding, we release BoigBench, a benchmark dataset containing 168 million enterprise tweets with their media, brand account names, time of post, and total likes.

We consider structured approximation of measures in Wasserstein space $W_p(\mathbb{R}^d)$ for $p\in[1,\infty)$ by discrete and piecewise constant measures based on a scaled Voronoi partition of $\mathbb{R}^d$. We show that if a full rank lattice $\Lambda$ is scaled by a factor of $h\in(0,1]$, then approximation of a measure based on the Voronoi partition of $h\Lambda$ is $O(h)$ regardless of $d$ or $p$. We then use a covering argument to show that $N$-term approximations of compactly supported measures is $O(N^{-\frac1d})$ which matches known rates for optimal quantizers and empirical measure approximation in most instances. Finally, we extend these results to noncompactly supported measures with sufficient decay.

We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

Integrating human feedback in models can improve the performance of natural language processing (NLP) models. Feedback can be either explicit (e.g. ranking used in training language models) or implicit (e.g. using human cognitive signals in the form of eyetracking). Prior eye tracking and NLP research reveal that cognitive processes, such as human scanpaths, gleaned from human gaze patterns aid in the understanding and performance of NLP models. However, the collection of real eyetracking data for NLP tasks is challenging due to the requirement of expensive and precise equipment coupled with privacy invasion issues. To address this challenge, we propose ScanTextGAN, a novel model for generating human scanpaths over text. We show that ScanTextGAN-generated scanpaths can approximate meaningful cognitive signals in human gaze patterns. We include synthetically generated scanpaths in four popular NLP tasks spanning six different datasets as proof of concept and show that the models augmented with generated scanpaths improve the performance of all downstream NLP tasks.

In this paper we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into $L^2$ spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a pre-specified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.