Dhagash Mehta

Yang-Hui He, Dhagash Mehta, Matthew Niemerg et al.

Solving large polynomial systems with coefficient parameters are ubiquitous and constitute an important class of problems. We demonstrate the computational power of two methods--a symbolic one called the Comprehensive Gröbner basis and a numerical one called the cheater's homotopy-applied to studying both potential energy landscapes and a variety of questions arising from geometry and phenomenology. Particular attention is paid to an example in flux compactification where important physical quantities such as the gravitino and moduli masses and the string coupling can be efficiently extracted.

Jonathan Hauenstein, Yang-Hui He, Dhagash Mehta

We propose a new computational method to understand the vacuum moduli space of (supersymmetric) field theories. By combining numerical algebraic geometry (NAG) and elimination theory, we develop a powerful, efficient, and parallelizable algorithm to extract important information such as the dimension, branch structure, Hilbert series and subsequent operator counting, as well as variation according to coupling constants and mass parameters. We illustrate this method on a host of examples from gauge theory, string theory, and algebraic geometry.

Heather A. Harrington, Dhagash Mehta, Helen M. Byrne et al.

Many systems in biology, physics and engineering can be described by systems of ordinary differential equation containing many parameters. When studying the dynamic behavior of these large, nonlinear systems, it is useful to identify and characterize the steady-state solutions as the model parameters vary, a technically challenging problem in a high-dimensional parameter landscape. Rather than simply determining the number and stability of steady-states at distinct points in parameter space, we decompose the parameter space into finitely many regions, the steady-state solutions being consistent within each distinct region. From a computational algebraic viewpoint, the boundary of these regions is contained in the discriminant locus. We develop global and local numerical algorithms for constructing the discriminant locus and classifying the parameter landscape. We showcase our numerical approaches by applying them to molecular and cell-network models.

Tianran Chen, Dhagash Mehta

A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions.Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations.

Daniel K. Molzahn, Matthew Niemerg, Dhagash Mehta et al.

The power flow equations model the steady-state relationship between the power injections and voltage phasors in an electric power system. By separating the real and imaginary components of the voltage phasors, the power flow equations can be formulated as a system of quadratic polynomials. Only the real solutions to these polynomial equations are physically meaningful. This paper focuses on the maximum number of real solutions to the power flow equations. An upper bound on the number of real power flow solutions commonly used in the literature is the maximum number of complex solutions. There exist two- and three-bus systems for which all complex solutions are real. It is an open question whether this is also the case for larger systems. This paper investigates four-bus systems using techniques from numerical algebraic geometry and conjectures a negative answer to this question. In particular, this paper studies lossless, four-bus systems composed of PV buses connected by lines with arbitrary susceptances. Computing the Galois group, which is degenerate, enables conversion of the problem of counting the number of real solutions to the power flow equations into counting the number of positive roots of a univariate sextic polynomial. From this analysis, it is conjectured that the system has at most 16 real solutions, which is strictly less than the maximum number of complex solutions, namely 20. We also provide explicit parameter values where this system has 16 real solutions so that the conjectured upper bound is achievable.

Dhruv Desai, Ashmita Dhiman, Tushar Sharma et al.

Mutual fund categorization has become a standard tool for the investment management industry and is extensively used by allocators for portfolio construction and manager selection, as well as by fund managers for peer analysis and competitive positioning. As a result, a (unintended) miscategorization or lack of precision can significantly impact allocation decisions and investment fund managers. Here, we aim to quantify the effect of miscategorization of funds utilizing a machine learning based approach. We formulate the problem of miscategorization of funds as a distance-based outlier detection problem, where the outliers are the data-points that are far from the rest of the data-points in the given feature space. We implement and employ a Random Forest (RF) based method of distance metric learning, and compute the so-called class-wise outlier measures for each data-point to identify outliers in the data. We test our implementation on various publicly available data sets, and then apply it to mutual fund data. We show that there is a strong relationship between the outlier measures of the funds and their future returns and discuss the implications of our findings.

Tianran Chen, Jakub Marecek, Dhagash Mehta et al.

We compare three formulations of stationary equations of the Kuramoto model as systems of polynomial equations. In the comparison, we present bounds on the numbers of real equilibria based on the work of Bernstein, Kushnirenko, and Khovanskii, and performance of methods for the optimisation over the set of equilibria based on the work of Lasserre, both of which could be of independent interest.

Souvik Chandra, Dhagash Mehta, Aranya Chakrabortty

The solution space of any set of power flow equations may contain different number of real-valued solutions. The boundaries that separate these regions are referred to as power flow solution space boundaries. Knowledge of these boundaries is important as they provide a measure for voltage stability. Traditionally, continuation based methods have been employed to compute these boundaries on the basis of initial guesses for the solution. However, with rapid growth of renewable energy sources these boundaries will be increasingly affected by variable parameters such as penetration levels, locations of the renewable sources, and voltage set-points, making it difficult to generate an initial guess that can guarantee all feasible solutions for the power flow problem. In this paper we solve this problem by applying a numerical polynomial homotopy based continuation method. The proposed method guarantees to find all solution boundaries within a given parameter space up to a chosen level of discretization, independent of any initial guess. Power system operators can use this computational tool conveniently to plan the penetration levels of renewable sources at different buses. We illustrate the proposed method through simulations on 3-bus and 10-bus power system examples with renewable generation.

Bhaskarjit Sarmah, Benika Hall, Rohan Rao et al.

Extraction and interpretation of intricate information from unstructured text data arising in financial applications, such as earnings call transcripts, present substantial challenges to large language models (LLMs) even using the current best practices to use Retrieval Augmented Generation (RAG) (referred to as VectorRAG techniques which utilize vector databases for information retrieval) due to challenges such as domain specific terminology and complex formats of the documents. We introduce a novel approach based on a combination, called HybridRAG, of the Knowledge Graphs (KGs) based RAG techniques (called GraphRAG) and VectorRAG techniques to enhance question-answer (Q&A) systems for information extraction from financial documents that is shown to be capable of generating accurate and contextually relevant answers. Using experiments on a set of financial earning call transcripts documents which come in the form of Q&A format, and hence provide a natural set of pairs of ground-truth Q&As, we show that HybridRAG which retrieves context from both vector database and KG outperforms both traditional VectorRAG and GraphRAG individually when evaluated at both the retrieval and generation stages in terms of retrieval accuracy and answer generation. The proposed technique has applications beyond the financial domain

Shaunak D. Bopardikar, Dhagash Mehta, Jonathan D. Hauenstein

We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate the use of tools from numerical algebraic geometry, in particular, a numerical polynomial homotopy continuation method that guarantees to find all solutions of polynomial equations, in order to characterize the \emph{global minima} for the coverage control problem. The results are then compared against a classic distributed approach involving the use of Lloyd descent, which is known to converge only to a local minimum under certain technical conditions.

Eren Kurshan, Dhagash Mehta, Bayan Bruss et al.

Adoption of AI by criminal entities across traditional and emerging financial crime paradigms has been a disturbing recent trend. Particularly concerning is the proliferation of generative AI, which has empowered criminal activities ranging from sophisticated phishing schemes to the creation of hard-to-detect deep fakes, and to advanced spoofing attacks to biometric authentication systems. The exploitation of AI by criminal purposes continues to escalate, presenting an unprecedented challenge. AI adoption causes an increasingly complex landscape of fraud typologies intertwined with cybersecurity vulnerabilities. Overall, GenAI has a transformative effect on financial crimes and fraud. According to some estimates, GenAI will quadruple the fraud losses by 2027 with a staggering annual growth rate of over 30% [27]. As crime patterns become more intricate, personalized, and elusive, deploying effective defensive AI strategies becomes indispensable. However, several challenges hinder the necessary progress of AI-based fincrime detection systems. This paper examines the latest trends in AI/ML-driven financial crimes and detection systems. It underscores the urgent need for developing agile AI defenses that can effectively counteract the rapidly emerging threats. It also aims to highlight the need for cooperation across the financial services industry to tackle the GenAI induced crime waves.

Guanchao Feng, Dhruv Desai, Stefano Pasquali et al.

In many real-world classification or recognition tasks, it is often difficult to collect training examples that exhaust all possible classes due to, for example, incomplete knowledge during training or ever changing regimes. Therefore, samples from unknown/novel classes may be encountered in testing/deployment. In such scenarios, the classifiers should be able to i) perform classification on known classes, and at the same time, ii) identify samples from unknown classes. This is known as open-set recognition. Although random forest has been an extremely successful framework as a general-purpose classification (and regression) method, in practice, it usually operates under the closed-set assumption and is not able to identify samples from new classes when run out of the box. In this work, we propose a novel approach to enabling open-set recognition capability for random forest classifiers by incorporating distance metric learning and distance-based open-set recognition. The proposed method is validated on both synthetic and real-world datasets. The experimental results indicate that the proposed approach outperforms state-of-the-art distance-based open-set recognition methods.

Mingshu Li, Bhaskarjit Sarmah, Dhruv Desai et al.

Due to the dynamic nature of financial markets, maintaining models that produce precise predictions over time is difficult. Often the goal isn't just point prediction but determining uncertainty. Quantifying uncertainty, especially the aleatoric uncertainty due to the unpredictable nature of market drivers, helps investors understand varying risk levels. Recently, quantile regression forests (QRF) have emerged as a promising solution: Unlike most basic quantile regression methods that need separate models for each quantile, quantile regression forests estimate the entire conditional distribution of the target variable with a single model, while retaining all the salient features of a typical random forest. We introduce a novel approach to compute quantile regressions from random forests that leverages the proximity (i.e., distance metric) learned by the model and infers the conditional distribution of the target variable. We evaluate the proposed methodology using publicly available datasets and then apply it towards the problem of forecasting the average daily volume of corporate bonds. We show that using quantile regression using Random Forest proximities demonstrates superior performance in approximating conditional target distributions and prediction intervals to the original version of QRF. We also demonstrate that the proposed framework is significantly more computationally efficient than traditional approaches to quantile regressions.

Bhaskarjit Sarmah, Tianjie Zhu, Dhagash Mehta et al.

For a financial analyst, the question and answer (Q\&A) segment of the company financial report is a crucial piece of information for various analysis and investment decisions. However, extracting valuable insights from the Q\&A section has posed considerable challenges as the conventional methods such as detailed reading and note-taking lack scalability and are susceptible to human errors, and Optical Character Recognition (OCR) and similar techniques encounter difficulties in accurately processing unstructured transcript text, often missing subtle linguistic nuances that drive investor decisions. Here, we demonstrate the utilization of Large Language Models (LLMs) to efficiently and rapidly extract information from earnings report transcripts while ensuring high accuracy transforming the extraction process as well as reducing hallucination by combining retrieval-augmented generation technique as well as metadata. We evaluate the outcomes of various LLMs with and without using our proposed approach based on various objective metrics for evaluating Q\&A systems, and empirically demonstrate superiority of our method.

Nathalia Castellanos, Dhruv Desai, Sebastian Frank et al.

Peer analysis is a critical component of investment management, often relying on expert-provided categorization systems. These systems' consistency is questioned when they do not align with cohorts from unsupervised clustering algorithms optimized for various metrics. We investigate whether unsupervised clustering can reproduce ground truth classes in a labeled dataset, showing that success depends on feature selection and the chosen distance metric. Using toy datasets and fund categorization as real-world examples we demonstrate that accurately reproducing ground truth classes is challenging. We also highlight the limitations of standard clustering evaluation metrics in identifying the optimal number of clusters relative to the ground truth classes. We then show that if appropriate features are available in the dataset, and a proper distance metric is known (e.g., using a supervised Random Forest-based distance metric learning method), then an unsupervised clustering can indeed reproduce the ground truth classes as distinct clusters.

Gregory Yampolsky, Dhruv Desai, Mingshu Li et al.

The explainability of black-box machine learning algorithms, commonly known as Explainable Artificial Intelligence (XAI), has become crucial for financial and other regulated industrial applications due to regulatory requirements and the need for transparency in business practices. Among the various paradigms of XAI, Explainable Case-Based Reasoning (XCBR) stands out as a pragmatic approach that elucidates the output of a model by referencing actual examples from the data used to train or test the model. Despite its potential, XCBR has been relatively underexplored for many algorithms such as tree-based models until recently. We start by observing that most XCBR methods are defined based on the distance metric learned by the algorithm. By utilizing a recently proposed technique to extract the distance metric learned by Random Forests (RFs), which is both geometry- and accuracy-preserving, we investigate various XCBR methods. These methods amount to identify special points from the training datasets, such as prototypes, critics, counter-factuals, and semi-factuals, to explain the predictions for a given query of the RF. We evaluate these special points using various evaluation metrics to assess their explanatory power and effectiveness.

Joshua Rosaler, Dhruv Desai, Bhaskarjit Sarmah et al.

We initiate a novel approach to explain the predictions and out of sample performance of random forest (RF) regression and classification models by exploiting the fact that any RF can be mathematically formulated as an adaptive weighted K nearest-neighbors model. Specifically, we employ a recent result that, for both regression and classification tasks, any RF prediction can be rewritten exactly as a weighted sum of the training targets, where the weights are RF proximities between the corresponding pairs of data points. We show that this linearity facilitates a local notion of explainability of RF predictions that generates attributions for any model prediction across observations in the training set, and thereby complements established feature-based methods like SHAP, which generate attributions for a model prediction across input features. We show how this proximity-based approach to explainability can be used in conjunction with SHAP to explain not just the model predictions, but also out-of-sample performance, in the sense that proximities furnish a novel means of assessing when a given model prediction is more or less likely to be correct. We demonstrate this approach in the modeling of US corporate bond prices and returns in both regression and classification cases.

Yaxuan Kong, Hoyoung Lee, Yoontae Hwang et al.

Large Language Models (LLMs) are increasingly integrated into financial workflows, but evaluation practice has not kept up. Finance-specific biases can inflate performance, contaminate backtests, and make reported results useless for any deployment claim. We identify five recurring biases in financial LLM applications. They include look-ahead bias, survivorship bias, narrative bias, objective bias, and cost bias. These biases break financial tasks in distinct ways and they often compound to create an illusion of validity. We reviewed 164 papers from 2023 to 2025 and found that no single bias is discussed in more than 28 percent of studies. This position paper argues that bias in financial LLM systems requires explicit attention and that structural validity should be enforced before any result is used to support a deployment claim. We propose a Structural Validity Framework and an evaluation checklist with minimal requirements for bias diagnosis and future system design. The material is available at https://github.com/Eleanorkong/Awesome-Financial-LLM-Bias-Mitigation.

Joshua S. Harvey, Guanchao Feng, Sai Anusha Meesala et al.

Despite their enormous predictive power, machine learning models are often unsuitable for applications in regulated industries such as finance, due to their limited capacity to provide explanations. While model-agnostic frameworks such as Shapley values have proved to be convenient and popular, they rarely align with the kinds of causal explanations that are typically sought after. Counterfactual case-based explanations, where an individual is informed of which circumstances would need to be different to cause a change in outcome, may be more intuitive and actionable. However, finding appropriate counterfactual cases is an open challenge, as is interpreting which features are most critical for the change in outcome. Here, we pose the question of counterfactual search and interpretation in terms of similarity learning, exploiting the representation learned by the random forest predictive model itself. Once a counterfactual is found, the feature importance of the explanation is computed as a function of which random forest partitions are crossed in order to reach it from the original instance. We demonstrate this method on both the MNIST hand-drawn digit dataset and the German credit dataset, finding that it generates explanations that are sparser and more useful than Shapley values.

Dimitris Vamvourellis, Dhagash Mehta

We investigate the effectiveness of large language models (LLMs), including reasoning-based and non-reasoning models, in performing zero-shot financial sentiment analysis. Using the Financial PhraseBank dataset annotated by domain experts, we evaluate how various LLMs and prompting strategies align with human-labeled sentiment in a financial context. We compare three proprietary LLMs (GPT-4o, GPT-4.1, o3-mini) under different prompting paradigms that simulate System 1 (fast and intuitive) or System 2 (slow and deliberate) thinking and benchmark them against two smaller models (FinBERT-Prosus, FinBERT-Tone) fine-tuned on financial sentiment analysis. Our findings suggest that reasoning, either through prompting or inherent model design, does not improve performance on this task. Surprisingly, the most accurate and human-aligned combination of model and method was GPT-4o without any Chain-of-Thought (CoT) prompting. We further explore how performance is impacted by linguistic complexity and annotation agreement levels, uncovering that reasoning may introduce overthinking, leading to suboptimal predictions. This suggests that for financial sentiment classification, fast, intuitive "System 1"-like thinking aligns more closely with human judgment compared to "System 2"-style slower, deliberative reasoning simulated by reasoning models or CoT prompting. Our results challenge the default assumption that more reasoning always leads to better LLM decisions, particularly in high-stakes financial applications.

Bhaskarjit Sarmah, Kriti Dutta, Anna Grigoryan et al.

We argue that the Declarative Self-improving Python (DSPy) optimizers are a way to align the large language model (LLM) prompts and their evaluations to the human annotations. We present a comparative analysis of five teleprompter algorithms, namely, Cooperative Prompt Optimization (COPRO), Multi-Stage Instruction Prompt Optimization (MIPRO), BootstrapFewShot, BootstrapFewShot with Optuna, and K-Nearest Neighbor Few Shot, within the DSPy framework with respect to their ability to align with human evaluations. As a concrete example, we focus on optimizing the prompt to align hallucination detection (using LLM as a judge) to human annotated ground truth labels for a publicly available benchmark dataset. Our experiments demonstrate that optimized prompts can outperform various benchmark methods to detect hallucination, and certain telemprompters outperform the others in at least these experiments.

Tianjiao Zhao, Jingrao Lyu, Stokes Jones et al.

The field of artificial intelligence (AI) agents is evolving rapidly, driven by the capabilities of Large Language Models (LLMs) to autonomously perform and refine tasks with human-like efficiency and adaptability. In this context, multi-agent collaboration has emerged as a promising approach, enabling multiple AI agents to work together to solve complex challenges. This study investigates the application of role-based multi-agent systems to support stock selection in equity research and portfolio management. We present a comprehensive analysis performed by a team of specialized agents and evaluate their stock-picking performance against established benchmarks under varying levels of risk tolerance. Furthermore, we examine the advantages and limitations of employing multi-agent frameworks in equity analysis, offering critical insights into their practical efficacy and implementation challenges.

Joshua S. Harvey, Joshua Rosaler, Mingshu Li et al.

We describe the use of an unsupervised Random Forest for similarity learning and improved unsupervised anomaly detection. By training a Random Forest to discriminate between real data and synthetic data sampled from a uniform distribution over the real data bounds, a distance measure is obtained that anisometrically transforms the data, expanding distances at the boundary of the data manifold. We show that using distances recovered from this transformation improves the accuracy of unsupervised anomaly detection, compared to other commonly used detectors, demonstrated over a large number of benchmark datasets. As well as improved performance, this method has advantages over other unsupervised anomaly detection methods, including minimal requirements for data preprocessing, native handling of missing data, and potential for visualizations. By relating outlier scores to partitions of the Random Forest, we develop a method for locally explainable anomaly predictions in terms of feature importance.

Bhaskarjit Sarmah, Mingshu Li, Jingrao Lyu et al.

To ensure and monitor large language models (LLMs) reliably, various evaluation metrics have been proposed in the literature. However, there is little research on prescribing a methodology to identify a robust threshold on these metrics even though there are many serious implications of an incorrect choice of the thresholds during deployment of the LLMs. Translating the traditional model risk management (MRM) guidelines within regulated industries such as the financial industry, we propose a step-by-step recipe for picking a threshold for a given LLM evaluation metric. We emphasize that such a methodology should start with identifying the risks of the LLM application under consideration and risk tolerance of the stakeholders. We then propose concrete and statistically rigorous procedures to determine a threshold for the given LLM evaluation metric using available ground-truth data. As a concrete example to demonstrate the proposed methodology at work, we employ it on the Faithfulness metric, as implemented in various publicly available libraries, using the publicly available HaluBench dataset. We also lay a foundation for creating systematic approaches to select thresholds, not only for LLMs but for any GenAI applications.

Samarth Sarin, Lovepreet Singh, Bhaskarjit Sarmah et al.

Agentic memory is emerging as a key enabler for large language models (LLM) to maintain continuity, personalization, and long-term context in extended user interactions, critical capabilities for deploying LLMs as truly interactive and adaptive agents. Agentic memory refers to the memory that provides an LLM with agent-like persistence: the ability to retain and act upon information across conversations, similar to how a human would. We present Memoria, a modular memory framework that augments LLM-based conversational systems with persistent, interpretable, and context-rich memory. Memoria integrates two complementary components: dynamic session-level summarization and a weighted knowledge graph (KG)-based user modelling engine that incrementally captures user traits, preferences, and behavioral patterns as structured entities and relationships. This hybrid architecture enables both short-term dialogue coherence and long-term personalization while operating within the token constraints of modern LLMs. We demonstrate how Memoria enables scalable, personalized conversational artificial intelligence (AI) by bridging the gap between stateless LLM interfaces and agentic memory systems, offering a practical solution for industry applications requiring adaptive and evolving user experiences.

Mingshu Li, Dhruv Desai, Jerinsh Jeyapaulraj et al.

Accurately measuring portfolio similarity is critical for a wide range of financial applications, including Exchange-traded Fund (ETF) recommendation, portfolio trading, and risk alignment. Existing similarity measures often rely on exact asset overlap or static distance metrics, which fail to capture similarities among the constituents (e.g., securities within the portfolio) as well as nuanced relationships between partially overlapping portfolios with heterogeneous weights. We introduce STRAPSim (Semantic, Two-level, Residual-Aware Portfolio Similarity), a novel method that computes portfolio similarity by matching constituents based on semantic similarity, weighting them according to their portfolio share, and aggregating results via residual-aware greedy alignment. We benchmark our approach against Jaccard, weighted Jaccard, as well as BERTScore-inspired variants across public classification, regression, and recommendation tasks, as well as on corporate bond ETF datasets. Empirical results show that our method consistently outperforms baselines in predictive accuracy and ranking alignment, achieving the highest Spearman correlation with return-based similarity. By leveraging constituent-aware matching and dynamic reweighting, portfolio similarity offers a scalable, interpretable framework for comparing structured asset baskets, demonstrating its utility in ETF benchmarking, portfolio construction, and systematic execution.

Chen Ling, Xujiang Zhao, Jiaying Lu et al.

Large language models (LLMs) have significantly advanced the field of natural language processing (NLP), providing a highly useful, task-agnostic foundation for a wide range of applications. However, directly applying LLMs to solve sophisticated problems in specific domains meets many hurdles, caused by the heterogeneity of domain data, the sophistication of domain knowledge, the uniqueness of domain objectives, and the diversity of the constraints (e.g., various social norms, cultural conformity, religious beliefs, and ethical standards in the domain applications). Domain specification techniques are key to make large language models disruptive in many applications. Specifically, to solve these hurdles, there has been a notable increase in research and practices conducted in recent years on the domain specialization of LLMs. This emerging field of study, with its substantial potential for impact, necessitates a comprehensive and systematic review to better summarize and guide ongoing work in this area. In this article, we present a comprehensive survey on domain specification techniques for large language models, an emerging direction critical for large language model applications. First, we propose a systematic taxonomy that categorizes the LLM domain-specialization techniques based on the accessibility to LLMs and summarizes the framework for all the subcategories as well as their relations and differences to each other. Second, we present an extensive taxonomy of critical application domains that can benefit dramatically from specialized LLMs, discussing their practical significance and open challenges. Last, we offer our insights into the current research status and future trends in this area.

Vipul Satone, Dhruv Desai, Dhagash Mehta

Identifying similar mutual funds with respect to the underlying portfolios has found many applications in financial services ranging from fund recommender systems, competitors analysis, portfolio analytics, marketing and sales, etc. The traditional methods are either qualitative, and hence prone to biases and often not reproducible, or, are known not to capture all the nuances (non-linearities) among the portfolios from the raw data. We propose a radically new approach to identify similar funds based on the weighted bipartite network representation of funds and their underlying assets data using a sophisticated machine learning method called Node2Vec which learns an embedded low-dimensional representation of the network. We call the embedding \emph{Fund2Vec}. Ours is the first ever study of the weighted bipartite network representation of the funds-assets network in its original form that identifies structural similarity among portfolios as opposed to merely portfolio overlaps.

Edgar A. Bernal, Jonathan D. Hauenstein, Dhagash Mehta et al.

Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classification boundaries over the parameter space, with the classes being the number of real solutions. For multidimensional parameter spaces, this article presents a novel sampling method which carefully samples the parameter space. At each sample point, homotopy continuation is used to obtain the number of real solutions to the corresponding polynomial system. Machine learning techniques including nearest neighbor and deep learning are used to efficiently approximate the real discriminant locus. One application of having learned the real discriminant locus is to develop a real homotopy method that only tracks the real solution paths unlike traditional methods which track all~complex~solution~paths. Examples show that the proposed approach can efficiently approximate complicated solution boundaries such as those arising from the equilibria of the Kuramoto model.

Dhagash Mehta, Dhruv Desai, Jithin Pradeep

Given the surge in popularity of mutual funds (including exchange-traded funds (ETFs)) as a diversified financial investment, a vast variety of mutual funds from various investment management firms and diversification strategies have become available in the market. Identifying similar mutual funds among such a wide landscape of mutual funds has become more important than ever because of many applications ranging from sales and marketing to portfolio replication, portfolio diversification and tax loss harvesting. The current best method is data-vendor provided categorization which usually relies on curation by human experts with the help of available data. In this work, we establish that an industry wide well-regarded categorization system is learnable using machine learning and largely reproducible, and in turn constructing a truly data-driven categorization. We discuss the intellectual challenges in learning this man-made system, our results and their implications.

Timothy E. Wang, Yiming Gu, Dhagash Mehta et al.

We investigate the topics of sensitivity and robustness in feedforward and convolutional neural networks. Combining energy landscape techniques developed in computational chemistry with tools drawn from formal methods, we produce empirical evidence indicating that networks corresponding to lower-lying minima in the optimization landscape of the learning objective tend to be more robust. The robustness estimate used is the inverse of a proposed sensitivity measure, which we define as the volume of an over-approximation of the reachable set of network outputs under all additive $l_{\infty}$-bounded perturbations on the input data. We present a novel loss function which includes a sensitivity term in addition to the traditional task-oriented and regularization terms. In our experiments on standard machine learning and computer vision datasets, we show that the proposed loss function leads to networks which reliably optimize the robustness measure as well as other related metrics of adversarial robustness without significant degradation in the classification error. Experimental results indicate that the proposed method outperforms state-of-the-art sensitivity-based learning approaches with regards to robustness to adversarial attacks. We also show that although the introduced framework does not explicitly enforce an adversarial loss, it achieves competitive overall performance relative to methods that do.

Dhagash Mehta, Tianran Chen, Tingting Tang et al.

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of the deep linear neural network models. After clarifying on the various definitions of "flat" minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized $L_2$ regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Using these upper bounds and utilizing a numerical algebraic geometry method, we find all stationary points of modest depth and matrix size. We show that in the presence of the non-zero regularization, deep linear networks indeed possess local minima which are not the global minima. Our computational results clarify certain aspects of the loss surfaces of deep linear networks and provide novel insights.

Dhagash Mehta, Xiaojun Zhao, Edgar A. Bernal et al.

Training an artificial neural network involves an optimization process over the landscape defined by the cost (loss) as a function of the network parameters. We explore these landscapes using optimisation tools developed for potential energy landscapes in molecular science. The number of local minima and transition states (saddle points of index one), as well as the ratio of transition states to minima, grow rapidly with the number of nodes in the network. There is also a strong dependence on the regularisation parameter, with the landscape becoming more convex (fewer minima) as the regularisation term increases. We demonstrate that in our formulation, stationary points for networks with $N_h$ hidden nodes, including the minimal network required to fit the XOR data, are also stationary points for networks with $N_{h} +1$ hidden nodes when all the weights involving the additional nodes are zero. Hence, smaller networks optimized to train the XOR data are embedded in the landscapes of larger networks. Our results clarify certain aspects of the classification and sensitivity (to perturbations in the input data) of minima and saddle points for this system, and may provide insight into dropout and network compression.

Andrew J. Ballard, Ritankar Das, Stefano Martiniani et al.

Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions. In particular, we can define quantities analogous to molecular structure, thermodynamics, and kinetics, and relate these emergent properties to the structure of the underlying landscape. This Perspective aims to describe these analogies with examples from recent applications, and suggest avenues for new interdisciplinary research.

Christian Knoll, Franz Pernkopf, Dhagash Mehta et al.

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.

Dhagash Mehta, Crina Grosan

Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.