Mohit Singh

Mohit Singh, Shehryar Khattak, Ashish Goel et al.

Visual-Inertial Odometry (VIO) provides smooth, high-rate state estimates and has been widely used for robotic navigation in both terrestrial and planetary applications. However, its performance is typically dependent on the frequency of visual updates, which is a challenge for planetary rovers operating under extreme resource constraints and low frame rates. This work investigates enabling reliable VIO with very sparse visual updates for lunar rover applications, addressing both day and night-time operations where feature associations become especially difficult under self-illumination conditions. We propose a Bird's Eye View (BEV)-based image matching scheme that remains robust to larger inter-frame motions and more reliable feature matching despite significant visual appearance changes. We extensively evaluate our proposed approach, BEVIO, through high-fidelity photorealistic lunar and real-time robotic experiments conducted using a half-scale lunar rover, in a long-term day-night deployment at Plaster City, CA, USA. The results demonstrate that our method enables reliable day and nighttime self-illuminated traverses at visual update rates as low as 0.25 Hz, underscoring its suitability for navigation on power- and compute-limited lunar rovers.

Mehrdad Ghadiri, Mohit Singh, Santosh S. Vempala

We study approximation algorithms for the socially fair $(\ell_p, k)$-clustering problem with $m$ groups, whose special cases include the socially fair $k$-median ($p=1$) and socially fair $k$-means ($p=2$) problems. We present (1) a polynomial-time $(5+2\sqrt{6})^p$-approximation with at most $k+m$ centers (2) a $(5+2\sqrt{6}+ε)^p$-approximation with $k$ centers in time $n^{2^{O(p)}\cdot m^2}$, and (3) a $(15+6\sqrt{6})^p$ approximation with $k$ centers in time $k^{m}\cdot\text{poly}(n)$. The first result is obtained via a refinement of the iterative rounding method using a sequence of linear programs. The latter two results are obtained by converting a solution with up to $k+m$ centers to one with $k$ centers using sparsification methods for (2) and via an exhaustive search for (3). We also compare the performance of our algorithms with existing bicriteria algorithms as well as exactly $k$ center approximation algorithms on benchmark datasets, and find that our algorithms also outperform existing methods in practice.

Morten Nissov, Mohit Singh, Kostas Alexis

Visual-Inertial Odometry (VIO) is a staple for reliable state estimation on constrained and lightweight platforms due to its versatility and demonstrated performance. However, pertinent challenges regarding robust operation in dark, low-texture, obscured environments complicate the use of such methods. Alternatively, Frequency Modulated Continuous Wave (FMCW) radars, and by extension Radar-Inertial Odometry (RIO), offer robustness to these visual challenges, albeit at the cost of reduced information density and worse long-term accuracy. To address these limitations, this work combines the two in a tightly coupled manner, enabling the resulting method to operate robustly regardless of environmental conditions or trajectory dynamics. The proposed method fuses image features, radar Doppler measurements, and Inertial Measurement Unit (IMU) measurements within an Iterated Extended Kalman Filter (IEKF) in real-time, with radar range data augmenting the visual feature depth initialization. The method is evaluated through flight experiments conducted in both indoor and outdoor environments, as well as through challenges to both exteroceptive modalities (such as darkness, fog, or fast flight), thoroughly demonstrating its robustness. The implementation of the proposed method is available at: https://github.com/ntnu-arl/radvio

Mohit Singh, Kostas Alexis

This paper presents a general refractive camera model and online co-estimation of odometry and the refractive index of unknown media. This enables operation in diverse and varying refractive fluids, given only the camera calibration in air. The refractive index is estimated online as a state variable of a monocular visual-inertial odometry framework in an iterative formulation using the proposed camera model. The method was verified on data collected using an underwater robot traversing inside a pool. The evaluations demonstrate convergence to the ideal refractive index for water despite significant perturbations in the initialization. Simultaneously, the approach enables on-par visual-inertial odometry performance in refractive media without prior knowledge of the refractive index or requirement of medium-specific camera calibration.

Swati Gupta, Jai Moondra, Mohit Singh

OMD and its variants give a flexible framework for OCO where the performance depends crucially on the choice of the mirror map. While the geometries underlying OPGD and OEG, both special cases of OMD, are well understood, it remains a challenging open question on how to construct an optimal mirror map for any given constrained set and a general family of loss functions, e.g., sparse losses. Motivated by parameterizing a near-optimal set of mirror maps, we consider a simpler question: is it even possible to obtain polynomial gains in regret by using mirror maps for geometries that interpolate between $L_1$ and $L_2$, which may not be possible by restricting to only OEG ($L_1$) or OPGD ($L_2$). Our main result answers this question positively. We show that mirror maps based on block norms adapt better to the sparsity of loss functions, compared to previous $L_p$ (for $p \in [1, 2]$) interpolations. In particular, we construct a family of online convex optimization instances in $\mathbb{R}^d$, where block norm-based mirror maps achieve a provable polynomial (in $d$) improvement in regret over OEG and OPGD for sparse loss functions. We then turn to the setting in which the sparsity level of the loss functions is unknown. In this case, the choice of geometry itself becomes an online decision problem. We first show that naively switching between OEG and OPGD can incur linear regret, highlighting the intrinsic difficulty of geometry selection. To overcome this issue, we propose a meta-algorithm based on multiplicative weights that dynamically selects among a family of uniform block norms. We show that this approach effectively tunes OMD to the sparsity of the losses, yielding adaptive regret guarantees. Overall, our results demonstrate that online mirror-map selection can significantly enhance the ability of OMD to exploit sparsity in online convex optimization.

Amit Ganz Rozenman, Ariel Kulik, Roy Schwartz et al.

We study the problem of maximizing a monotone submodular function subject to a matroid independence constraint. For more than a decade, a rich body of work has studied this problem. Initially, a tight approximation of $ (1-\frac{1}{e})$ was given using the continuous greedy algorithm [Calinescu-Chekuri-Pal-Vondr{á}k STOC`2008] and later non-oblivious local search techniques were able to match this tight approximation guarantee [Filmus-Ward FOCS`2012] and [Buchbinder-Feldman FOCS`2024]. We propose a new and remarkably simple approach to this problem that is based on a stochastic Poisson process. Our approach matches the tight $ (1-\frac{1}{e})$ approximation guarantee and it differs from the known two techniques since it does not require discretization or rounding while performing very few single element swaps. We also present applications of our approach and obtain fast algorithms for submodular welfare maximization, and for the general and separable assignment problems.

Rohan Ghuge, Jai Moondra, Mohit Singh

We study the Stochastic Boolean Function Certification (SBFC) problem, where we are given $n$ Bernoulli random variables $\{X_e: e \in U\}$ on a ground set $U$ of $n$ elements with joint distribution $p$, a Boolean function $f: 2^U \to \{0, 1\}$, and an (unknown) scenario $S = \{e \in U: X_e = 1\}$ of active elements sampled from $p$. We seek to probe the elements one-at-a-time to reveal if they are active until we can certify $f(S) = 1$, while minimizing the expected number of probes. Unlike most previous results that assume independence, we study correlated distributions $p$ and give approximation algorithms for several classes of functions $f$. When $f(S)$ is the indicator function for whether $S$ is the spanning set of a given matroid, our problem reduces to finding a basis of active elements of a matroid by probing elements. We give a non-adaptive $O(\log n)$-approximation algorithm for arbitrary distributions $p$, and show that this is tight up to constants unless P $=$ NP, even for partition matroids. For uniform matroids, we give constant factor $4.642$-approximation ([BBFT20]) that can be further improved to a $2$-approximation if additionally the random variables are negatively correlated for the case of $1$-uniform matroid. We also give an adaptive $O(\log k)$-approximation algorithm for SBFC for $k$-uniform matroids for the Graph Probing problem, where we seek to probe the edges of a graph one-at-a-time until we find $k$ active edges. The underlying distribution on edges arises from (hidden) independent vertex random variables, with an edge being active if at least one of its endpoints is active. This significantly improves over the information-theoretic lower bound on $Ω(\mathrm{poly}(n))$ ([JGM19]) for adaptive algorithms for $k$-uniform matroids with arbitrary distributions.

Vivek Madan, Aleksandar Nikolov, Mohit Singh et al.

Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning. The current best results include an $e^{2k}$-estimation for any matroid of rank $k$ and a $(1+ε)^d$-approximation for a uniform matroid of rank $k\ge d+\frac dε$, where the rank $k\ge d$ denotes the desired size of the optimal set. Our main result is a new approximation algorithm with an approximation guarantee that depends only on the dimension $d$ of the vectors and not on the size $k$ of the output set. In particular, we show an $(O(d))^{d}$-estimation and an $(O(d))^{d^3}$-approximation for any matroid, giving a significant improvement over prior work when $k\gg d$. Our result relies on the existence of an optimal solution to a convex programming relaxation for the problem which has sparse support; in particular, no more than $O(d^2)$ variables of the solution have fractional values. The sparsity results rely on the interplay between the first-order optimality conditions for the convex program and matroid theory. We believe that the techniques introduced to show sparsity of optimal solutions to convex programs will be of independent interest. We also give a randomized algorithm that rounds a sparse fractional solution to a feasible integral solution to the original problem. To show the approximation guarantee, we utilize recent works on strongly log-concave polynomials and show new relationships between different convex programs studied for the problem. Finally, we use the estimation algorithm and sparsity results to give an efficient deterministic approximation algorithm with an approximation guarantee that depends solely on the dimension $d$.

Uthaipon Tantipongpipat, Samira Samadi, Mohit Singh et al.

Dimensionality reduction is a classical technique widely used for data analysis. One foundational instantiation is Principal Component Analysis (PCA), which minimizes the average reconstruction error. In this paper, we introduce the "multi-criteria dimensionality reduction" problem where we are given multiple objectives that need to be optimized simultaneously. As an application, our model captures several fairness criteria for dimensionality reduction such as our novel Fair-PCA problem and the Nash Social Welfare (NSW) problem. In Fair-PCA, the input data is divided into $k$ groups, and the goal is to find a single $d$-dimensional representation for all groups for which the minimum variance of any one group is maximized. In NSW, the goal is to maximize the product of the individual variances of the groups achieved by the common low-dimensional space. Our main result is an exact polynomial-time algorithm for the two-criterion dimensionality reduction problem when the two criteria are increasing concave functions. As an application of this result, we obtain a polynomial time algorithm for Fair-PCA for $k=2$ groups and a polynomial time algorithm for NSW objective for $k=2$ groups. We also give approximation algorithms for $k>2$. Our technical contribution in the above results is to prove new low-rank properties of extreme point solutions to semi-definite programs. We conclude with experiments indicating the effectiveness of algorithms based on extreme point solutions of semi-definite programs on several real-world data sets.

Samira Samadi, Uthaipon Tantipongpipat, Jamie Morgenstern et al.

We investigate whether the standard dimensionality reduction technique of PCA inadvertently produces data representations with different fidelity for two different populations. We show on several real-world data sets, PCA has higher reconstruction error on population A than on B (for example, women versus men or lower- versus higher-educated individuals). This can happen even when the data set has a similar number of samples from A and B. This motivates our study of dimensionality reduction techniques which maintain similar fidelity for A and B. We define the notion of Fair PCA and give a polynomial-time algorithm for finding a low dimensional representation of the data which is nearly-optimal with respect to this measure. Finally, we show on real-world data sets that our algorithm can be used to efficiently generate a fair low dimensional representation of the data.

Sebastian Pokutta, Mohit Singh, Alfredo Torrico

In this work, we consider robust submodular maximization with matroid constraints. We give an efficient bi-criteria approximation algorithm that outputs a small family of feasible sets whose union has (nearly) optimal objective value. This algorithm theoretically performs less function calls than previous works at cost of adding more elements to the final solution. We also provide significant implementation improvements showing that our algorithm outperforms the algorithms in the existing literature. We finally assess the performance of our contributions in three real-world applications.

Mohit Singh, Weijun Xie

Experimental design is a classical statistics problem and its aim is to estimate an unknown $m$-dimensional vector $β$ from linear measurements where a Gaussian noise is introduced in each measurement. For the combinatorial experimental design problem, the goal is to pick $k$ out of the given $n$ experiments so as to make the most accurate estimate of the unknown parameters, denoted as $\hatβ$. In this paper, we will study one of the most robust measures of error estimation - $D$-optimality criterion, which corresponds to minimizing the volume of the confidence ellipsoid for the estimation error $β-\hatβ$. The problem gives rise to two natural variants depending on whether repetitions of experiments are allowed or not. We first propose an approximation algorithm with a $\frac1e$-approximation for the $D$-optimal design problem with and without repetitions, giving the first constant factor approximation for the problem. We then analyze another sampling approximation algorithm and prove that it is $(1-ε)$-approximation if $k\geq \frac{4m}ε+\frac{12}{ε^2}\log(\frac{1}ε)$ for any $ε\in (0,1)$. Finally, for $D$-optimal design with repetitions, we study a different algorithm proposed by literature and show that it can improve this asymptotic approximation ratio.

Aleksandar Nikolov, Mohit Singh, Uthaipon Tao Tantipongpipat

We study the optimal design problems where the goal is to choose a set of linear measurements to obtain the most accurate estimate of an unknown vector in $d$ dimensions. We study the $A$-optimal design variant where the objective is to minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks, sparse least squares regression, feature selection for $k$-means clustering, and matrix approximation. In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for $A$-optimal design. Our main result is to obtain improved approximation algorithms for the $A$-optimal design problem by introducing the proportional volume sampling algorithm. Our results nearly optimal bounds in the asymptotic regime when the number of measurements done, $k$, is significantly more than the dimension $d$. We also give first approximation algorithms when $k$ is small including when $k=d$. The proportional volume-sampling algorithm also gives approximation algorithms for other optimal design objectives such as $D$-optimal design and generalized ratio objective matching or improving previous best known results. Interestingly, we show that a similar guarantee cannot be obtained for the $E$-optimal design problem. We also show that the $A$-optimal design problem is NP-hard to approximate within a fixed constant when $k=d$.

Sébastien Bubeck, Yin Tat Lee, Mohit Singh

We propose a new method for unconstrained optimization of a smooth and strongly convex function, which attains the optimal rate of convergence of Nesterov's accelerated gradient descent. The new algorithm has a simple geometric interpretation, loosely inspired by the ellipsoid method. We provide some numerical evidence that the new method can be superior to Nesterov's accelerated gradient descent.