arXiv subjects

Vishal Gupta

Publications and source records attributed to Vishal Gupta.

At least 19 recordsLinked to original sources

arXiv2026

Edge complexity of graphs

Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.

math.CO↗

arXiv2026

Fourier Ratios of Graph Kernels: Energy Bounds, Optimal Labelings, and Recovery

We study labeling-sensitive Fourier complexity for finite graph kernels. After identifying the vertices of a graph with the cyclic group $\mathbb Z_N$, its adjacency matrix becomes a function on $\mathbb Z_N^2$. Minimizing the quotient of the $\ell^1$ and $\ell^2$ norms of its two-dimensional Fourier transform over all vertex labelings gives an isomorphism invariant $\operatorname{FR}_{\min}(G)$. A nuclear-norm argument gives \[\operatorname{FR}_{\min}(G) \geq \frac{\mathcal E(G)}{\sqrt{2s}},\] where $s$ is the number of edges and $\mathcal E(G)$ is the graph energy. The natural cyclic labeling attains equality for every circulant graph. We obtain exact formulas for several graph families and a labeling-sensitive complete bipartite example. We also connect the invariant with the Fourier algebra of $\mathbb Z_N^2$. The quantitative Cohen idempotent theorem implies that every Boolean kernel of bounded Fourier ratio has an exact signed coset decomposition whose length is independent of $N$. A previously established Fourier-ratio recovery theorem gives stable Frobenius approximation of a fixed labeled adjacency matrix from Bernoulli samples. We distinguish this conclusion from exact edge recovery and from the problem of finding a good labeling. For a Laplacian eigenvalue of multiplicity $m(\lambda)$, we prove \[ \operatorname{FR}_{\min}(\Pi_\lambda) \geq \sqrt{m(\lambda)}, \] with equality for circulant graphs. Strongly regular graphs and the Petersen graph show how adjacency and projector complexity can agree or differ. Direct projector sampling yields heat-kernel approximation. We conclude with a graph-signal spectral synthesis principle and asymptotic uniqueness from incomplete vertex data.

math.CO↗

arXiv2026

Large values in time series and additive combinatorics

It is well-known in industrial data science that large values of real-life time series tend to be structured and often follow concrete and visible patterns. In this paper, we use ideas from additive combinatorics and discrete Fourier analysis to give this heuristic a mathematical foundation. Our main tool is the Fourier ratio, a complexity measure previously used in compressed sensing, combined with a generalized version of Chang's lemma from additive combinatorics. Together, these yield a precise prediction: when the Fourier ratio of a time series is small, the set of its largest values can be additively generated by a very small set using only $\{-1,0,1\}$ coefficients. We test this prediction on US inflation data and Delhi climate data, both in their original form and after mean-centering. The numerical results confirm the predicted structure: a generating set of size $4$--$7$ suffices to span large spectra containing dozens of points, even when the Fourier ratio is large enough that our theoretical bounds become loose. These findings provide a rigorous explanation for why extreme values in real-world data are information-rich and structurally significant.

math.CO↗

arXiv2025

The non-existence of some Moore polygons and spectral Moore bounds

In this paper, we study the maximum order $v(k,\theta)$ of a connected $k$-regular graph whose second largest eigenvalue is at most $\theta$. From Alon-Boppana and Serre, we know that $v(k,\theta)$ is finite when $\theta < 2\sqrt{k-1}$ while the work of Marcus, Spielman, and Srivastava implies that $v(k,\theta)$ is infinite if $\theta\geq 2\sqrt{k-1}$. Cioab\u{a}, Koolen, Nozaki, and Vermette obtained a general upper bound on $v(k, \theta)$ via Nozaki's linear programming bound and determined many values of $v(k,\theta)$. The graphs attaining this bound are distance-regular and are called Moore polygons. Damerell and Georgiacodis proved that there are no Moore polygons of diameter $6$ or more. For smaller diameters, there are infinitely many Moore polygons. We complement these results by proving two nonexistence results for Moore polygons with specific parameters. We also determine new values of $v(k,\theta)$: $v(4, \sqrt{2}) = 14$ and $v(5, \sqrt{2}) = v(5,\sqrt{5}-1)=16$. The former is achieved by the co-Heawood graph, and the latter by the folded $5$-cube. We verify that any connected $5$-regular graph with second eigenvalue $\lambda_2$ exceeding $1$ satisfies $\lambda_2 \geq \sqrt{5} - 1$, and that the unique $5$-regular graph attaining equality in this bound has $10$ vertices. We prove a stronger form of a 2015 conjecture of Kolokolnikov related to the second eigenvalue of cubic graphs of given order, and observe that other recent results on the second eigenvalue of regular graphs are consequences of the general upper bound theorem on $v(k,\theta)$ mentioned above.

math.CO↗

arXiv2025

Minimum spectral radius of graphs of fixed order and dissociation number and its connection to Tur\'an problems

Let $\mathcal{D}_{n,\tau}$ be the set of all simple connected graphs of order $n$ and dissociation number $\tau.$ In this paper, we study the minimum size and the minimum spectral radius of graphs in $\mathcal{D}_{n,\tau}$ in connection with Tur\'an-type problems for complete multipartite graphs. We characterize the Tur\' an graphs for several complete multipartite graphs where the size of one of the partite sets is much smaller than the size of the remaining partites. This extends a result of Erd\H{o}s and Simonovits [16]. Additionally, we prove some stability results to get the structure of graphs without such a forbidden complete multipartite subgraph, and close to Tur\'an number of edges. As an application, we show that a graph with the minimum spectral radius in $\mathcal{D}_{n,\tau}$ must be a graph with the minimum size in $\mathcal{D}_{n, \tau}$ when $n$ is sufficiently large and satisfies some parity conditions. We then describe a few structural properties of graphs with the minimum spectral radius in $\mathcal{D}_{n,\tau}$. For even dissociation numbers and any order $n$, we compute the minimum size of a graph in $\mathcal{D}_{n,\tau}$ and use it to characterize the graphs in $\mathcal{D}_{n, 4}$ that attain the minimum size and the minimum spectral radius. We also apply the stability results to upper bound the minimum number of edges and spectral radius for connected graphs with a given $d$-independence number when the order of the graph is sufficiently large. Finally, we derive two new bounds on the value of $\tau(G)$ for a given graph $G$.

math.CO↗

arXiv2025

Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions

It is proved that \[ \sum_{\chi \bmod q}N(\sigma,T,\chi) \ll_{\epsilon} (qT)^{7(1-\sigma)/3+\epsilon}, \] where $N(\sigma,T,\chi)$ denotes the number of zeros $\rho=\beta+it$ of $L(s,\chi)$ in the rectangle $\sigma\leq \beta\leq 1$, $|t|\leq T$. The exponent $7/3$ improves upon Huxley's earlier exponent of $12/5$. The key innovation lies in deriving a sharp upper bound for sums over affine transformations of functions with a GCD twist, which arises from our adaptation of the Guth--Maynard method. As applications of the zero density estimates obtained in this paper, we derive a new upper bound for the least Goldbach number in arithmetic progressions modulo a prime and establish new results on primes in arithmetic progressions in short intervals, in particular for prime-power moduli.

math.NT↗

arXiv2025

agriFrame: Agricultural framework to remotely control a rover inside a greenhouse environment

The growing demand for innovation in agriculture is essential for food security worldwide and more implicit in developing countries. With growing demand comes a reduction in rapid development time. Data collection and analysis are essential in agriculture. However, considering a given crop, its cycle comes once a year, and researchers must wait a few months before collecting more data for the given crop. To overcome this hurdle, researchers are venturing into digital twins for agriculture. Toward this effort, we present an agricultural framework(agriFrame). Here, we introduce a simulated greenhouse environment for testing and controlling a robot and remotely controlling/implementing the algorithms in the real-world greenhouse setup. This work showcases the importance/interdependence of network setup, remotely controllable rover, and messaging protocol. The sophisticated yet simple-to-use agriFrame has been optimized for the simulator on minimal laptop/desktop specifications.

cs.RO↗

arXiv2024

Beyond Discretization: Learning the Optimal Solution Path

Many applications require minimizing a family of optimization problems indexed by some hyperparameter $\lambda \in \Lambda$ to obtain an entire solution path. Traditional approaches proceed by discretizing $\Lambda$ and solving a series of optimization problems. We propose an alternative approach that parameterizes the solution path with a set of basis functions and solves a \emph{single} stochastic optimization problem to learn the entire solution path. Our method offers substantial complexity improvements over discretization. When using constant-step size SGD, the uniform error of our learned solution path relative to the true path exhibits linear convergence to a constant related to the expressiveness of the basis. When the true solution path lies in the span of the basis, this constant is zero. We also prove stronger results for special cases common in machine learning: When $\lambda \in [-1, 1]$ and the solution path is $\nu$-times differentiable, constant step-size SGD learns a path with $\epsilon$ uniform error after at most $O(\epsilon^{\frac{1}{1-\nu}} \log(1/\epsilon))$ iterations, and when the solution path is analytic, it only requires $O\left(\log^2(1/\epsilon)\log\log(1/\epsilon)\right)$. By comparison, the best-known discretization schemes in these settings require at least $O(\epsilon^{-1/2})$ discretization points (and even more gradient calls). Finally, we propose an adaptive variant of our method that sequentially adds basis functions and demonstrates strong numerical performance through experiments.

math.OC↗

arXiv2024

On the minimum spectral radius of connected graphs of given order and size

In this paper, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices $n$ and number of edges $e$, the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this paper, we answer Hong's question positively for various values of $n$ and $e$ and in several cases, we determined the graphs with minimum spectral radius.

math.CO↗

arXiv2024

Node resistance curvature in Cartesian graph products

Devriendt and Lambiotte recently introduced the \emph{node resistance curvature}, a notion of graph curvature based on the effective resistance matrix. In this paper, we begin the study of the behavior of the node resistance curvature under the operation of the Cartesian graph product. We study the natural question of global positivity of node resistance curvature of the Cartesian product of positively-curved graphs, and prove that, whenever $m,n\ge3$, the node resistance curvature of the interior vertices of a $m\times n$ grid is always nonpositive, while it is always nonnegative on the boundary of such grids. For completeness, we also prove a number of results on node resistance curvature in $2\times n$ grids and exhibit a counterexample to a generalization. We also give generic bounds and suggest several further questions for future study.

math.CO↗

arXiv2024

A Ricci flow on graphs from effective resistance

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci--Foster curvature. We study the Ricci flow resulting from this curvature. We prove the existence of solutions to Ricci flow on short time intervals, and prove that Ricci flow preserves graphs with nonnegative (resp. positive) curvature.

math.CO↗

arXiv2024

Decision-Focused Learning with Directional Gradients

We propose a novel family of decision-aware surrogate losses, called Perturbation Gradient (PG) losses, for the predict-then-optimize framework. The key idea is to connect the expected downstream decision loss with the directional derivative of a particular plug-in objective, and then approximate this derivative using zeroth order gradient techniques. Unlike the original decision loss which is typically piecewise constant and discontinuous, our new PG losses is a Lipschitz continuous, difference of concave functions that can be optimized using off-the-shelf gradient-based methods. Most importantly, unlike existing surrogate losses, the approximation error of our PG losses vanishes as the number of samples grows. Hence, optimizing our surrogate loss yields a best-in-class policy asymptotically, even in misspecified settings. This is the first such result in misspecified settings, and we provide numerical evidence confirming our PG losses substantively outperform existing proposals when the underlying model is misspecified.

cs.LG↗

arXiv2023

Balanced Off-Policy Evaluation for Personalized Pricing

We consider a personalized pricing problem in which we have data consisting of feature information, historical pricing decisions, and binary realized demand. The goal is to perform off-policy evaluation for a new personalized pricing policy that maps features to prices. Methods based on inverse propensity weighting (including doubly robust methods) for off-policy evaluation may perform poorly when the logging policy has little exploration or is deterministic, which is common in pricing applications. Building on the balanced policy evaluation framework of Kallus (2018), we propose a new approach tailored to pricing applications. The key idea is to compute an estimate that minimizes the worst-case mean squared error or maximizes a worst-case lower bound on policy performance, where in both cases the worst-case is taken with respect to a set of possible revenue functions. We establish theoretical convergence guarantees and empirically demonstrate the advantage of our approach using a real-world pricing dataset.

stat.ML↗

arXiv2022

A lower bound for the smallest eigenvalue of a graph and an application to the associahedron graph

In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply our results to obtain a lower bound on the smallest eigenvalue of the associahedron graph, and we prove that this bound gives the correct order of magnitude of this eigenvalue. We also survey what is known regarding the second-largest eigenvalue of the associahedron graph.

math.CO↗

arXiv2022

Debiasing In-Sample Policy Performance for Small-Data, Large-Scale Optimization

Motivated by the poor performance of cross-validation in settings where data are scarce, we propose a novel estimator of the out-of-sample performance of a policy in data-driven optimization.Our approach exploits the optimization problem's sensitivity analysis to estimate the gradient of the optimal objective value with respect to the amount of noise in the data and uses the estimated gradient to debias the policy's in-sample performance. Unlike cross-validation techniques, our approach avoids sacrificing data for a test set, utilizes all data when training and, hence, is well-suited to settings where data are scarce. We prove bounds on the bias and variance of our estimator for optimization problems with uncertain linear objectives but known, potentially non-convex, feasible regions. For more specialized optimization problems where the feasible region is "weakly-coupled" in a certain sense, we prove stronger results. Specifically, we provide explicit high-probability bounds on the error of our estimator that hold uniformly over a policy class and depends on the problem's dimension and policy class's complexity. Our bounds show that under mild conditions, the error of our estimator vanishes as the dimension of the optimization problem grows, even if the amount of available data remains small and constant. Said differently, we prove our estimator performs well in the small-data, large-scale regime. Finally, we numerically compare our proposed method to state-of-the-art approaches through a case-study on dispatching emergency medical response services using real data. Our method provides more accurate estimates of out-of-sample performance and learns better-performing policies.

math.OC↗

arXiv2021

ETA Prediction with Graph Neural Networks in Google Maps

Travel-time prediction constitutes a task of high importance in transportation networks, with web mapping services like Google Maps regularly serving vast quantities of travel time queries from users and enterprises alike. Further, such a task requires accounting for complex spatiotemporal interactions (modelling both the topological properties of the road network and anticipating events -- such as rush hours -- that may occur in the future). Hence, it is an ideal target for graph representation learning at scale. Here we present a graph neural network estimator for estimated time of arrival (ETA) which we have deployed in production at Google Maps. While our main architecture consists of standard GNN building blocks, we further detail the usage of training schedule methods such as MetaGradients in order to make our model robust and production-ready. We also provide prescriptive studies: ablating on various architectural decisions and training regimes, and qualitative analyses on real-world situations where our model provides a competitive edge. Our GNN proved powerful when deployed, significantly reducing negative ETA outcomes in several regions compared to the previous production baseline (40+% in cities like Sydney).

cs.LG↗

arXiv2021

Micro BTB: A High Performance and Lightweight Last-Level Branch Target Buffer for Servers

High-performance branch target buffers (BTBs) and the L1I cache are key to high-performance front-end. Modern branch predictors are highly accurate, but with an increase in code footprint in modern-day server workloads, BTB and L1I misses are still frequent. Recent industry trend shows usage of large BTBs (100s of KB per core) that provide performance closer to the ideal BTB along with a decoupled front-end that provides efficient fetch-directed L1I instruction prefetching. On the other hand, techniques proposed by academia, like BTB prefetching and using retire order stream for learning, fail to provide significant performance with modern-day processor cores that are deeper and wider. We solve the problem fundamentally by increasing the storage density of the last-level BTB. We observe that not all branch instructions require a full branch target address. Instead, we can store the branch target as a branch offset, relative to the branch instruction. Using branch offset enables the BTB to store multiple branches per entry. We reduce the BTB storage in half, but we observe that it increases skewness in the BTB. We propose a skewed indexed and compressed last-level BTB design called MicroBTB (MBTB) that stores multiple branches per BTB entry. We evaluate MBTB on 100 industry-provided server workloads. A 4K-entry MBTB provides 17.61% performance improvement compared to an 8K-entry baseline BTB design with a storage savings of 47.5KB per core.

cs.AR↗

arXiv2020

Balance Regularized Neural Network Models for Causal Effect Estimation

Estimating individual and average treatment effects from observational data is an important problem in many domains such as healthcare and e-commerce. In this paper, we advocate balance regularization of multi-head neural network architectures. Our work is motivated by representation learning techniques to reduce differences between treated and untreated distributions that potentially arise due to confounding factors. We further regularize the model by encouraging it to predict control outcomes for individuals in the treatment group that are similar to control outcomes in the control group. We empirically study the bias-variance trade-off between different weightings of the regularizers, as well as between inductive and transductive inference.

cs.LG↗