Bala Krishnamoorthy

Shreyas Subramanian, Bala Krishnamoorthy, Pranav Murthy

Despite significant advances in optimizers for training, most research works use common scheduler choices like Cosine or exponential decay. In this paper, we study \emph{GreedyLR}, a novel scheduler that adaptively adjusts the learning rate during training based on the current loss. To validate the effectiveness of our proposed scheduler, we conduct experiments on several NLP, CV, and LLM tasks with up to $7B$ parameters, including both fine-tuning and pre-training experiments. The results show that our approach outperforms several state-of-the-art schedulers in terms of accuracy, speed, and convergence. We also provide a theoretical analysis of the GreedyLR algorithm, including a proof of convergence and derivation of the optimal scaling factor $F$ that maximizes the convergence rate, along with experiments to show robustness of the algorithm to realistic noisy landscapes. Our scheduler is easy to implement, computationally efficient, and could be considered a good default scheduler for training.

Nathan H. May, Bala Krishnamoorthy, Patrick Gambill

Persistence diagrams (PDs) are used as signatures of point cloud data. Two clouds of points can be compared using the bottleneck distance d_B between their PDs. A potential drawback of this pipeline is that point clouds sampled from topologically similar manifolds can have arbitrarily large d_B when there is a large scaling between them. This situation is typical in dimension reduction frameworks. We define, and study properties of, a new scale-invariant distance between PDs termed normalized bottleneck distance, d_N. In defining d_N, we develop a broader framework called metric decomposition for comparing finite metric spaces of equal cardinality with a bijection. We utilize metric decomposition to prove a stability result for d_N by deriving an explicit bound on the distortion of the bijective map. We then study two popular dimension reduction techniques, Johnson-Lindenstrauss (JL) projections and metric multidimensional scaling (mMDS), and a third class of general biLipschitz mappings. We provide new bounds on how well these dimension reduction techniques preserve homology with respect to d_N. For a JL map f that transforms input X to f(X), we show that d_N(dgm(X),dgm(f(X))) < e, where dgm(X) is the Vietoris-Rips PD of X, and pairwise distances are preserved by f up to the tolerance 0 < ε< 1. For mMDS, we present new bounds for d_B and d_N between PDs of X and its projection in terms of the eigenvalues of the covariance matrix. And for k-biLipschitz maps, we show that d_N is bounded by the product of (k^2-1)/k and the ratio of diameters of X and f(X). Finally, we use computational experiments to demonstrate the increased effectiveness of using the normalized bottleneck distance for clustering sets of point clouds sampled from different shapes.

Youjia Zhou, Methun Kamruzzaman, Patrick Schnable et al.

High-throughput technologies to collect field data have made observations possible at scale in several branches of life sciences. The data collected can range from the molecular level (genotypes) to physiological (phenotypic traits) and environmental observations (e.g., weather, soil conditions). These vast swathes of data, collectively referred to as phenomics data, represent a treasure trove of key scientific knowledge on the dynamics of the underlying biological system. However, extracting information and insights from these complex datasets remains a significant challenge owing to their multidimensionality and lack of prior knowledge about their complex structure. In this paper, we present Pheno-Mapper, an interactive toolbox for the exploratory analysis and visualization of large-scale phenomics data. Our approach uses the mapper framework to perform a topological analysis of the data, and subsequently render visual representations with built-in data analysis and machine learning capabilities. We demonstrate the utility of this new tool on real-world plant (e.g., maize) phenomics datasets. In comparison to existing approaches, the main advantage of Pheno-Mapper is that it provides rich, interactive capabilities in the exploratory analysis of phenomics data, and it integrates visual analytics with data analysis and machine learning in an easily extensible way. In particular, Pheno-Mapper allows the interactive selection of subpopulations guided by a topological summary of the data and applies data mining and machine learning to these selected subpopulations for in-depth exploration.

Youjia Zhou, Nathaniel Saul, Ilkin Safarli et al.

The mapper construction is a powerful tool from topological data analysis that is designed for the analysis and visualization of multivariate data. In this paper, we investigate a method for stitching a pair of univariate mappers together into a bivariate mapper, and study topological notions of information gains, referred to as topological gains, during such a process. We further provide implementations that visualize such topological gains for mapper graphs.

Prashant Gupta, Bala Krishnamoorthy, Gregory Dreifus

We develop a framework that creates a new polygonal mesh representation of the sparse infill domain of a layer-by-layer 3D printing job. We guarantee the existence of a single, continuous tool path covering each connected piece of the domain in every layer. We present a tool path algorithm that traverses each such continuous tool path with no crossovers. The key construction at the heart of our framework is an Euler transformation which converts a 2-dimensional cell complex K into a new 2-complex K^ such that every vertex in the 1-skeleton G^ of K^ has even degree. Hence G^ is Eulerian, and a Eulerian tour can be followed to print all edges in a continuous fashion. We start with a mesh K of the union of polygons obtained by projecting all layers to the plane. We compute its Euler transformation K^. In the slicing step, we clip K^ at each layer using its polygon to obtain a complex that may not necessarily be Euler. We then patch this complex by adding edges such that any odd-degree nodes created by slicing are transformed to have even degrees again. We print extra support edges in place of any segments left out to ensure there are no edges without support in the next layer. These support edges maintain the Euler nature of the complex. Finally we describe a tree-based search algorithm that builds the continuous tool path by traversing "concentric" cycles in the Euler complex. Our algorithm produces a tool path that avoids material collisions and crossovers, and can be printed in a continuous fashion irrespective of complex geometry or topology of the domain (e.g., holes). We implement our test our framework on several 3D objects. Apart from standard geometric shapes, we demonstrate the framework on the Stanford bunny.

Dustin L. Arendt, Matthew Broussard, Bala Krishnamoorthy et al.

We define a new filtration called the Steinhaus filtration built from a single cover based on a generalized Steinhaus distance, a generalization of Jaccard distance. The homology persistence module of a Steinhaus filtration with infinitely many cover elements may not be $q$-tame, even when the covers are in a totally bounded space. While this may pose a challenge to derive stability results, we show that the Steinhaus filtration is stable when the cover is finite. We show that while the Čech and Steinhaus filtrations are not isomorphic in general, they are isomorphic for a finite point set in dimension one. Furthermore, the VR filtration completely determines the $1$-skeleton of the Steinhaus filtration in arbitrary dimension. We then develop a language and theory for stable paths within the Steinhaus filtration. We demonstrate how the framework can be applied to several applications where a standard metric may not be defined but a cover is readily available. We introduce a new perspective for modeling recommendation system datasets. As an example, we look at a movies dataset and we find the stable paths identified in our framework represent a sequence of movies constituting a gentle transition and ordering from one genre to another. For explainable machine learning, we apply the Mapper algorithm for model induction by building a filtration from a single Mapper complex, and provide explanations in the form of stable paths between subpopulations. For illustration, we build a Mapper complex from a supervised machine learning model trained on the FashionMNIST dataset. Stable paths in the Steinhaus filtration provide improved explanations of relationships between subpopulations of images.

Nathan Hamlin, Bala Krishnamoorthy, William Webb

We had recently shown that every positive integer can be represented uniquely using a recurrence sequence, when certain restrictions on the digit strings are satisfied. We present the details of how such representations can be used to build a knapsack-like public key cryptosystem. We also present new disguising methods, and provide arguments for the security of the code against known methods of attack.