arXiv subjects

Arijit Ghosh

Publications and source records attributed to Arijit Ghosh.

At least 19 recordsLinked to original sources

arXiv2026

Power of Conditional Sampling on Hypergraphs

Let $G=(V,E)$ be a $d$-uniform hypergraph with $|V|=n$ and $|E|=m$. Estimating $m$ from uniformly random hyperedge samples requires $Θ(\sqrt{m})$ samples by a birthday-paradox argument. We investigate the power of conditional sampling through a hierarchy of oracles: for an integer $1\le s\le d$, given pairwise disjoint subsets $X_1,\ldots,X_s\subseteq V$, the $s$-colorful oracle returns a hyperedge chosen uniformly at random from all hyperedges contained in $X_1\cup\cdots\cup X_s$ that intersect every $X_i$, provided one exists. For $s=1$, this samples a uniformly random hyperedge induced by $X_1$. For every $1\le s\le d$, we give a randomized algorithm that outputs $\widehat{m}\in[(1-\varepsilon)m,(1+\varepsilon)m]$ with probability at least $2/3$ using $O_{s,d}(\operatorname{poly}(1/\varepsilon,\log n))$ conditional samples, where the exponent of $\log n$ is an absolute constant independent of both $s$ and $d$. Conditional sampling thus yields an exponential separation over uniform hyperedge sampling. We also study the decision version, in which the oracle only reports whether a qualifying hyperedge exists. For every $1\le s<d$, we prove that any algorithm using the $s$-colorful decision oracle requires $\widetildeΩ_{s,d}(n^{\frac{d-s}{d-s+1}})$ queries. Hence, for $s<d$, access to uniformly random witness hyperedges (random certificates) is exponentially more powerful than existence queries. For $s=d$, decision queries alone achieve $\operatorname{poly}(1/\varepsilon,\log n)$ query complexity, but the exponent of $\log n$ is known to be $Θ(d)$. The colorful hierarchy thus exhibits a phase transition at $s=d$: for $s<d$, random certificates are essential for polylogarithmic-query estimation, whereas at $s=d$ decision queries suffice, yet random certificates still provably reduce the exponent of $\log n$ from $Θ(d)$ to an absolute constant.

cs.DS↗

arXiv2026

Constructive Arithmetic Regularity for Abelian Groups

The arithmetic regularity lemma of Green is a fundamental tool in additive combinatorics, providing a decomposition of arbitrary subsets of finite Abelian groups into pseudorandom pieces with respect to Fourier analysis. Over the years, it has found numerous applications in combinatorics, property testing, and theoretical computer science. However, Green's proof is existential, and many algorithmic applications require an efficient procedure for constructing the underlying regular decomposition. De, Nadimpalli, and Servedio (SODA 2022) developed a query-efficient constructive regularity framework over $\mathbb{F}_2^n$ and explicitly raised the problem of extending their dimension-independent sumset-estimation algorithm to other finite Abelian groups. In this work, we resolve this question for finite Abelian groups of constant torsion. We develop the first constructive version of Green's {\em Arithmetic Regularity Lemma} in this general setting. Given oracle access to a subset of the group, our algorithm computes an implicit description of a regular decomposition together with an oracle representation of a structured subset satisfying the guarantees of Green's lemma. The algorithm runs in time and query complexity polynomial in the regularity parameters and the tower bound, while remaining independent of the size of the ambient group. As applications, we obtain efficient algorithms for removing linear configurations and approximately counting linear images, including weighted sumsets and suitable matrix systems of Cauchy--Schwarz complexity one, over finite Abelian groups of constant torsion. Our results provide a unified algorithmic framework extending dimension-independent constructive arithmetic regularity and its applications beyond $\mathbb{F}_2^n$.

cs.DS↗

arXiv2026

Balancing Frequencies and Pixels in Flow Matching

Natural images follow a $1/f^2$ spectral distribution: most signal energy lies in the low spatial frequencies, while the perceptually important structures such as textures and edges occupy sparse high-frequency bands. Pixel-space reconstruction objectives, however, treat all spatial errors uniformly, causing low frequencies to dominate the optimization signal and delaying the learning of fine-scale details. In this work, we identify this objective-level spectral imbalance as a key inefficiency in training pixel-space flow models. To address it, we propose a Focal Log-Frequency Loss (f-loss), a spectrally balanced objective that equalizes the learning signal across frequencies, emphasizing high-frequency components that are otherwise underrepresented in pixel-space objectives. Building on this, we introduce a simple training strategy that combines frequency and pixel supervision: we first emphasize frequency-domain learning early to capture all frequencies, and then transition to standard pixel-space v-loss for spatial refinement. This balancing mitigates the low-frequency bias of pixel losses and aligns the training signal with the evolving needs of the model. Our approach is conceptually simple, requires no architectural changes, and acts as a drop-in replacement for flow matching losses. Across multiple model scales, it accelerates convergence by up to 40% while consistently improving FID and perceptual fidelity.

cs.CV↗

arXiv2026

Dino Forcing Flow Models: Do not denoise what you can predict

Co-denoising pretrained representations such as DINO can substantially improve the training speed and quality of flow matching models, but it introduces a second denoising trajectory and requires carefully designed schedules. We propose a simpler alternative: predict the pretrained representation directly, then condition the model on its own prediction. This removes the need for a second ODE and any representation-specific denoising schedules, while retaining the benefits of representation guidance. Our approach converges substantially faster and achieves better generation quality as measured by FID score. On ImageNet, it outperforms the state of the art in latent space at 2x fewer epochs than prior methods; in pixel space, it improves FID over comparable prior methods by more than 20%. These results support a simple principle: do not denoise what you can predict. Our code is openly available at https://github.com/arijit-hub/dino_forcing.

cs.CV↗

arXiv2026

Counting Triangles in Graph Streams with Repeatable and Forgettable Edges

Most existing graph streaming algorithms assume the ideal scenario where each edge arrives only once. Real-world graph streams, such as communication or transaction logs, often contain many repeated occurrences of the same edge. In general, the algorithms developed for the single-edge arrival case can fail when edges can arrive multiple times. Motivated by this, we study the {\em repeated-edge arrival graph streaming model} where an edge is allowed to arrive multiple times. In this work, we study the triangle counting problem in the repeated-edge arrival model: approximate the number of triangles in the underlying {\em simple graph} despite arbitrary edge repetitions. We design the first algorithms for triangle counting with optimal space complexity. In particular, we present a single-pass algorithm that computes an $(\varepsilon,δ)$-approximation of the number of triangles with optimal space complexity. We introduce {\em right-to-be-forgotten graph streaming} (RFGS) model, where a forget operation can cause all previous occurrences of an edge to disappear. We show that our single-pass algorithm can be extended to the RFGS model with optimal space complexity. Finally, we present optimal constant-pass algorithms that compute an $(\varepsilon,δ)$-approximation of the number of triangles and cliques for the repeated-edge arrival graph streams.

cs.DS↗

arXiv2026

Unlocking Fractional Moments in Delphic Set Streams

We consider estimation of non-integer frequency moments $F_k$ and related Bernstein-type statistics in the Delphic set stream model under a bounded-frequency assumption: every universe element appears at most $τ$ times. The main challenge of this model is to keep space low while also keeping update time low, which is not trivial because the sets can be exponential in size compared to their representations. Our core insight is that by sampling the stream at different rates and observing the resulting distinct-counts, we can 'probe' the frequency distribution and numerically integrate these probes to reconstruct a broad class of statistics. Building on that, we crucially observe that the distinct-count of a randomly sampled substream, viewed as a function of the sampling rate, is a single analytic object whose evaluations determine a broad class of statistics via a complementary Laplace-type integral. Algorithmically we exploit this by: 1. estimating those evaluations using only standard $F_0$ (distinct-count) algorithms on sampled substreams and 2. recovering target statistics by controlled numerical integration on a judiciously chosen grid. For $F_k$ with $k\in (0,1)$ we obtain the first one-pass streaming algorithms for Delphic set streams whose space and per-set update time are $\mathrm{poly}(\log|Ω|,\log m,\varepsilon^{-1},\log(1/δ))$ in the practically relevant regime $τ=\mathrm{polylog}(|Ω|,m)$; in general the bounds are polynomial in $τ$ and $\varepsilon^{-1}$ and logarithmic in $δ^{-1}$. We also give a complexity-theoretic barrier explaining why lower bounds for removing the bounded-frequency assumption appear difficult: ruling out polylogarithmic algorithms for unrestricted Delphic $F_k$ would imply a linear-space threshold-counting separation.

cs.DS↗

arXiv2026

From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries

We study the problem of estimating the number of edges in an undirected, unweighted graph using sublinear query access. We consider a query model that preserves the structure of Independent Set (IS) queries, but augments their output with a random certificate: given a vertex subset, the oracle returns a uniformly random edge from the induced subgraph if one exists, and returns null otherwise. Using this access, we give a randomized algorithm that outputs a $(1 \pm \varepsilon)$-approximation to the number of edges with constant success probability using $\widetilde{O}(\log^{2} m)$ queries. This implies an exponential separation from both standard IS queries and global random edge-sampling models: estimating the number of edges using standard IS queries require $\widetildeΘ\!\left(\min\left\{\sqrt{m},\, \frac{n}{\sqrt{m}}\right\}\right)$ queries, while direct random edge-sample access requires $\widetildeΘ(\sqrt{m})$ samples. Beyond separation in query complexity, our algorithm is output-sensitive: its query complexity is polylogarithmic in the number of edges in the graph. This aligns with the classical objective in group testing, where one seeks algorithms that are both worst-case optimal and instance-adaptive. Conceptually, our model connects group testing, the decision-versus-counting dichotomy, graph property testing, and the "power of a random certificate", and can be viewed as a structured form of conditional sampling of edges in graphs.

cs.DS↗

arXiv2026

Simplicial subdivision of simplices of arbitrary dimension in spaces of constant curvature with bounded quality

In 1942, Freudenthal showed that a simplex in Euclidean space can be subdivided such that the quality (well-shapedness of the simplex, quantified in terms of e.g. fatness) of the simplices in the subdivision is lower bounded. This answered a question of Brouwer. Recently, Brunck discussed the same problem for simplices in two-dimensional spaces of constant curvature and provided a closely related construction. In this paper we generalize Brunck's result to arbitrary dimensional spaces of constant curvature by combining Freudenthal's construction and radial projection. We contrast this approach with Brunck's construction.

cs.CG↗

arXiv2026

Almost covering all the layers of hypercube with multiplicities

Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of $\mathcal{Q}^{n}_{k}$. In this short note, we show that $$deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_{1}, \dots, H_{m}$ in $\mathbb{R}^{n}$, where $m = \max\left\{ k, n-k\right\}+2t-2$, such that every point of $\mathcal{Q}^{n}_{k}$ will be covered exactly $t-1$ times, and every other point of $\mathcal{Q}^{n}$ will be covered at least $t$ times. Note that putting $k = 0$ and $t=1$, we recover the much celebrated covering result of Alon and Füredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube $\mathcal{Q}^{n}$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.

math.CO↗

arXiv2026

RE-TRIANGLE: Does TRIANGLE Enable Multimodal Alignment Beyond Cosine Similarity in Retrieval?

Multimodal alignment is critical for bridging the semantic gap in information retrieval. However, traditional pairwise strategies introduce a geometric blind spot: while they align anchor modalities (e.g., text) with others, they lack constraints to enforce mutual consistency between peripheral modalities (e.g., video and audio). The TRIANGLE framework addresses this by minimizing the area of modality triplets on a hypersphere to enforce holistic alignment. In this reproducibility study, we verify the robustness of this geometric objective for retrieval tasks. We confirm that TRIANGLE outperforms pairwise baselines in zero-shot settings, achieving Recall@1 gains of up to +8.7 points, though benefits are domain-dependent. However, we fail to reproduce the reported learning-from-scratch results. Analysis using a synthetic toy dataset attributes this to instability when jointly optimizing geometric alignment with Data-Text Matching (DTM) loss. Furthermore, we find that cosine regularization primarily stabilizes text-to-video retrieval, and fine-tuning with domain supervision amplifies geometric benefits but reduces cross-dataset generalization. Our findings support the efficacy of geometric alignment while highlighting critical optimization sensitivities. Code available at https://github.com/ARIJIT00171/RE-TRIANGLE.

cs.IR↗

arXiv2026

MIRO: MultI-Reward cOnditioned pretraining improves T2I quality and efficiency

The default paradigm of post-training text-to-image generators includes post-hoc selection of generated images, and subsequent training with one reward model to align the generator to the reward, typically user preference. This discards informative data as well as optimizes only for a single reward, hence harming diversity, semantic fidelity and efficiency. Instead, we propose MIRO, a method that conditions the model on multiple rewards during training, thus letting the model learn user preferences directly. MIRO pre-training both improves the visual quality of the generated images and speeds up the training, achieving state of the art on the GenEval compositional benchmark and user-preference scores (PickAScore, ImageReward, HPSv2).

cs.CV↗

arXiv2026

PoM: A Linear-Time Replacement for Attention with the Polynomial Mixer

This paper introduces the Polynomial Mixer (PoM), a novel token mixing mechanism with linear complexity that serves as a drop-in replacement for self-attention. PoM aggregates input tokens into a compact representation through a learned polynomial function, from which each token retrieves contextual information. We prove that PoM satisfies the contextual mapping property, ensuring that transformers equipped with PoM remain universal sequence-to-sequence approximators. We replace standard self-attention with PoM across five diverse domains: text generation, handwritten text recognition, image generation, 3D modeling, and Earth observation. PoM matches the performance of attention-based models while drastically reducing computational cost when working with long sequences. The code is available at https://github.com/davidpicard/pom.

cs.CV↗

arXiv2026

Structure of sparse Boolean functions over Abelian groups, and its application to testing

We study Fourier-sparse Boolean functions over general finite Abelian groups. A Boolean function $f : G \to \{-1,+1\}$ is $s$-sparse if it has at most $s$ non-zero Fourier coefficients. We introduce a general notion of granularity of Fourier coefficients and prove that every non-zero coefficient of an $s$-sparse Boolean function has magnitude at least \begin{equation*} \frac{1}{2^{φ(Δ)/2} \, s^{φ(Δ)/2}}, \end{equation*} where $Δ$ denotes the exponent of the group $G$ (that is, the maximum order of an element in $G$) and $φ$ is the Euler's totient function. This generalizes the celebrated result of Gopalan et al. (SICOMP 2011) for $\mathbb{Z}_2^n$, extending it to all finite Abelian groups via new techniques from group theory and algebraic number theory. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient sparsity testing algorithm for Boolean functions. The tester distinguishes whether a given function is $s$-sparse or $ε$-far from every $s$-sparse Boolean function, with query complexity $poly\left((2s)^{φ(Δ)},1/ε\right)$. In addition, we generalize the classical notion of Boolean degree to arbitrary Abelian groups and establish an $Ω(\sqrt{s})$ lower bound for adaptive sparsity testing.

cs.CC↗

arXiv2026

Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions

Given Boolean functions $ f, g : \mathbb{F}_2^n \to \{-1,+1\} $, we say they are {\em linearly isomorphic} if there exists $ A \in \mathrm{GL}_n(\mathbb{F}_2) $ such that $ f(x)=g(Ax) $ for all $ x $. We study this problem in the tolerant property testing framework under the known--unknown model, where $ g $ is given explicitly and $ f $ is accessible only via oracle queries, meaning the algorithm may adaptively request the value of $ f(x) $ for inputs $ x \in \mathbb{F}_2^n $ of its choice. Given parameters $ ε\ge 0 $ and $ ω>0 $, the goal is to distinguish whether there exists $ A \in \mathrm{GL}_n(\mathbb{F}_{2})$ such that the normalized Hamming distance between $ f $ and $ g(Ax) $ is at most $ ε$, or whether for every $ A \in \mathrm{GL}_n(\mathbb{F}_2) $ the distance is at least $ ε+ω$. Our main result is a tolerant tester making $ \widetilde{O} \left( \left( m/ω\right)^4 \right) $ queries to $ f $, where $ m $ is an upper bound on the spectral norm of $ g $, improving the previous $ \widetilde{O} \left( \left( m/ω\right)^{24} \right) $ bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of $ Ω(m^2) $ for constant $ ω$ (for example, $ ω=1/4 $), improving the prior $ Ω(\log m) $ lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana--McFarland functions from symmetric-key cryptography.

cs.CC↗

arXiv2026

Spectral Shadows: When Communication Complexity Meets Linear Invariance Testing

In this short note, we initiate the study of the Linear Isomorphism Testing Problem in the setting of communication complexity, a natural linear algebraic generalization of the classical Equality problem. Given Boolean functions $f, g : \mathbb{F}_2^n \to \{-1, +1\}$, Alice and Bob are tasked with determining whether $f$ and $g$ are equivalent up to a nonsingular linear transformation of the input variables, or far from being so. This problem has been extensively investigated in several models of computation, including standard algorithmic and property testing frameworks, owing to its fundamental connections with combinatorial circuit design, complexity theory, and cryptography. However, despite its broad relevance, it has remained unexplored in the context of communication complexity, a gap we address in this work. Our main results demonstrate that the approximate spectral norm of the input functions plays a central role in governing the communication complexity of this problem. We design a simple deterministic protocol whose communication cost is polynomial in the approximate spectral norm, and complement it with nearly matching lower bounds (up to a quadratic gap). In the randomised setting with private coins, we present an even more efficient protocol, though equally simple, that achieves a quadratically improved dependence on the approximate spectral norm compared to the deterministic case, and we prove that such a dependence is essentially unavoidable. These results identify the approximate spectral norm as a key complexity measure for testing linear invariance in the communication complexity framework. As a core technical ingredient, we establish new junta theorems for Boolean functions with small approximate spectral norm, which may be of independent interest in Fourier analysis and learning theory.

cs.DS↗

arXiv2025

Colorful two-piercing theorem for boxes

We prove a colorful extension of a Helly-type theorem by Danzer and Grünbaum (Combinatorica, 1982) concerning two-piercing families of axis-parallel boxes in $\mathbb{R}^d$. We also show that our result is tight by constructing extremal families that achieve the bound. Related work includes a graph-theoretic proof of the original theorem by Pendavingh, Puite, and Woeginger (Discrete Applied Mathematics, 2008), and a two-piercing result for lower-dimensional boxes by Baños and Oliveros (Acta Mathematica Hungarica, 2018).

cs.CG↗

arXiv2025

Optimal non-adaptive algorithm for edge estimation

We present a simple nonadaptive randomized algorithm that estimates the number of edges in a simple, unweighted, undirected graph, possibly containing isolated vertices, using only degree and random edge queries. For an $n$-vertex graph, our method requires only $\widetilde{O}(\sqrt{n})$ queries, achieving sublinear query complexity. The algorithm independently samples a set of vertices and queries their degrees, and also independently samples a set of edges, using the answers to these queries to estimate the total number of edges in the graph. We further prove a matching lower bound, establishing the optimality of our algorithm and resolving the non-adaptive query complexity of this problem with respect to degree and random-edge queries.

cs.DS↗

arXiv2025

Helly theorem for affine spaces without dimensions

We prove a no-dimensional Helly theorem for affine spaces and convex sets using the unboundedness framework of Aronov, Goodman, and Pollack (Computational Geometry, 2002). This generalizes the fundamental result of Adiprasito, Bárány, Mustafa, and Terpai on the no-dimensional Helly theorem for points and convex sets (Discrete & Computational Geometry, 2020). Additionally, we establish the optimality of our result.

math.CO↗