arXiv subjects

Tommaso d'Orsi

Publications and source records attributed to Tommaso d'Orsi.

At least 19 recordsLinked to original sources

arXiv2026

Strongly Refuting Random CSP without Literals

Under what condition is a random constraint satisfaction problem hard to refute by the sum-of-squares (SoS) algorithm? A sufficient condition is t-wise uniformity, that is, each constraint has a t-wise uniform distribution of satisfying assignments, as shown by the lower bounds of Kothari, Mori, O'Donnell, and Witmer (STOC 2017). This condition is also necessary for random CSPs given by a predicate and uniformly random literals, due to the constant-degree SoS refutation of Allen, O'Donnell, and Witmer (FOCS 2015). For higher degree, Raghavendra, Rao, and Schramm (STOC 2017) gave a refutation for Boolean random CSPs with uniformly random literals, matching the lower bounds optimally in terms of the three-way tradeoff between constraint density, SoS degree, and strength of refutation. Two long-standing open problems are to find a more general sufficient condition for SoS lower bounds, and to refute similar random CSPs not involving literals. We show that for a general random k-CSP, the necessary and sufficient hardness condition is not t-wise uniformity, but t-wise independence. We generalize the optimal three-way tradeoff to any random k-CSP, without assuming a Boolean domain or uniformly random literals. Our analysis involves new Kikuchi matrices for odd order and for asymmetric tensors. It also uses the global correlation rounding technique of Barak, Raghavendra, and Steurer (FOCS 2011). To avoid the running-time penalty of this technique, we also give a spectral refutation algorithm.

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arXiv2026

Combinatorial Optimization using Comparison Oracles

In linear combinatorial optimization, we aim to find $S^* = \arg\min_{S \in \mathcal{F}} \langle w,\mathbf{1}_S \rangle$ for a family $\mathcal{F} \subseteq 2^U$ over a ground set $U$ of $n$ elements. Traditionally, $w$ is known or accessible via a value oracle. Motivated by practical applications involving pairwise preferences, we study the weaker and more robust comparison oracle, which for any $S, T \in \mathcal{F}$ reveals only if $w(S) <, =, \text{ or } > w(T)$. We investigate the query complexity and computational efficiency of optimizing in this model. We present three main contributions. (1) Query Complexity: We establish that the query complexity over any arbitrary set system $\mathcal{F} \subseteq 2^U$ is $\tilde{O}(n^2)$. This demonstrates a fundamental separation between information and computational complexity, as the runtime may still be exponential for NP-hard problems. (2) Algorithmic Frameworks: We develop two general tools. First, a Dual Ellipsoid framework establishes an efficient reduction from optimization to certification. It shows that to optimize efficiently, it suffices to efficiently certify a candidate's optimality using only comparisons. Second, Global Subspace Learning (GSL) sorts all feasible sets using $O(nB \log(nB))$ queries for integer weights bounded by $B$. We efficiently implement GSL for linear matroids, yielding improved query complexities for problems like $k$-SUM, SUBSET-SUM, and $A+B$ sorting. (3) Combinatorial Applications: We give the first polynomial-time, low-query algorithms for classic problems, including minimum cuts, minimum weight spanning trees (and matroid bases), bipartite matching (and matroid intersection), and shortest $s$-$t$ paths. Our work provides the first general query complexity bounds and efficient algorithmic results for this fundamental model.

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arXiv2026

On Purely Private Covariance Estimation

We present a simple perturbation mechanism for the release of $d$-dimensional covariance matrices $Σ$ under pure differential privacy. For large datasets with at least $n\geq d^2/\varepsilon$ elements, our mechanism recovers the provably optimal Frobenius norm error guarantees of \cite{nikolov2023private}, while simultaneously achieving best known error for all other $p$-Schatten norms, with $p\in [1,\infty]$. Our error is information-theoretically optimal for all $p\ge 2$, in particular, our mechanism is the first purely private covariance estimator that achieves optimal error in spectral norm. For small datasets $n< d^2/\varepsilon$, we further show that by projecting the output onto the nuclear norm ball of appropriate radius, our algorithm achieves the optimal Frobenius norm error $O(\sqrt{d\;\text{Tr}(Σ) /n})$, improving over the known bounds of $O(\sqrt{d/n})$ of \cite{nikolov2023private} and ${O}\big(d^{3/4}\sqrt{\text{Tr}(Σ)/n}\big)$ of \cite{dong2022differentially}.

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arXiv2025

Complexity of Local Search for CSPs Parameterized by Constraint Difference

In this paper, we study the parameterized complexity of local search, whose goal is to find a good nearby solution from the given current solution. Formally, given an optimization problem where the goal is to find the largest feasible subset $S$ of a universe $U$, the new input consists of a current solution $P$ (not necessarily feasible) as well as an ordinary input for the problem. Given the existence of a feasible solution $S^*$, the goal is to find a feasible solution as good as $S^*$ in parameterized time $f(k) \cdot n^{O(1)}$, where $k$ denotes the distance $|PΔS^*|$. This model generalizes numerous classical parameterized optimization problems whose parameter $k$ is the minimum number of elements removed from $U$ to make it feasible, which corresponds to the case $P = U$. We apply this model to widely studied Constraint Satisfaction Problems (CSPs), where $U$ is the set of constraints, and a subset $U'$ of constraints is feasible if there is an assignment to the variables satisfying all constraints in $U'$. We give a complete characterization of the parameterized complexity of all boolean-alphabet symmetric CSPs, where the predicate's acceptance depends on the number of true literals.

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arXiv2025

Tight Differentially Private PCA via Matrix Coherence

We revisit the task of computing the span of the top $r$ singular vectors $u_1, \ldots, u_r$ of a matrix under differential privacy. We show that a simple and efficient algorithm -- based on singular value decomposition and standard perturbation mechanisms -- returns a private rank-$r$ approximation whose error depends only on the \emph{rank-$r$ coherence} of $u_1, \ldots, u_r$ and the spectral gap $σ_r - σ_{r+1}$. This resolves a question posed by Hardt and Roth~\cite{hardt2013beyond}. Our estimator outperforms the state of the art -- significantly so in some regimes. In particular, we show that in the dense setting, it achieves the same guarantees for single-spike PCA in the Wishart model as those attained by optimal non-private algorithms, whereas prior private algorithms failed to do so. In addition, we prove that (rank-$r$) coherence does not increase under Gaussian perturbations. This implies that any estimator based on the Gaussian mechanism -- including ours -- preserves the coherence of the input. We conjecture that similar behavior holds for other structured models, including planted problems in graphs. We also explore applications of coherence to graph problems. In particular, we present a differentially private algorithm for Max-Cut and other constraint satisfaction problems under low coherence assumptions.

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arXiv2025

Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs

Whether or not the Sparsest Cut problem admits an efficient $O(1)$-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture. Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension $\text{SD}_\varepsilon(G)$: the smallest $k$ such that the subspace spanned by the first $k$ Laplacian eigenvectors contains all but $\varepsilon$ fraction of a sparsest cut. Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups -- canonical examples of graphs with poor expansion properties. We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a $(1+\varepsilon)$-approximation to the uniform Sparsest Cut of a degree-$d$ Cayley graph over an Abelian group of size $n$ in time $n^{O(1)}\cdot\exp(d/\varepsilon)^{O(d)}$. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of $O(1)$-approximate sparsest cuts has an $\varepsilon$-net of size $\exp(d/\varepsilon)^{O(d)}$ and that the multiplicity of $λ_2$ is bounded by $2^{O(d)}$. The latter bound is tight and improves on a previous bound of $2^{O(d^2)}$ by Lee and Makarychev.

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arXiv2025

On Finding Randomly Planted Cliques in Arbitrary Graphs

We study a planted clique model introduced by Feige where a complete graph of size $c\cdot n$ is planted uniformly at random in an arbitrary $n$-vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size $(c/3)^{O(1/c)} \cdot n$ as long as the original graph has maximum degree at most $(1-p)n$ for some fixed $p>0$. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical $G(n,\frac{1}{2})+K_{\sqrt{n}}$ planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size $Ω(n)$ for every fixed $c>0$, even if the input graph has maximum degree $(1-p)n$. Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.

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arXiv2024

Perturb-and-Project: Differentially Private Similarities and Marginals

We revisit the input perturbations framework for differential privacy where noise is added to the input $A\in \mathcal{S}$ and the result is then projected back to the space of admissible datasets $\mathcal{S}$. Through this framework, we first design novel efficient algorithms to privately release pair-wise cosine similarities. Second, we derive a novel algorithm to compute $k$-way marginal queries over $n$ features. Prior work could achieve comparable guarantees only for $k$ even. Furthermore, we extend our results to $t$-sparse datasets, where our efficient algorithms yields novel, stronger guarantees whenever $t\le n^{5/6}/\log n\,.$ Finally, we provide a theoretical perspective on why \textit{fast} input perturbation algorithms works well in practice. The key technical ingredients behind our results are tight sum-of-squares certificates upper bounding the Gaussian complexity of sets of solutions.

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arXiv2024

A Near-Linear Time Approximation Algorithm for Beyond-Worst-Case Graph Clustering

We consider the semi-random graph model of [Makarychev, Makarychev and Vijayaraghavan, STOC'12], where, given a random bipartite graph with $α$ edges and an unknown bipartition $(A, B)$ of the vertex set, an adversary can add arbitrary edges inside each community and remove arbitrary edges from the cut $(A, B)$ (i.e. all adversarial changes are \textit{monotone} with respect to the bipartition). For this model, a polynomial time algorithm is known to approximate the Balanced Cut problem up to value $O(α)$ [MMV'12] as long as the cut $(A, B)$ has size $Ω(α)$. However, it consists of slow subroutines requiring optimal solutions for logarithmically many semidefinite programs. We study the fine-grained complexity of the problem and present the first near-linear time algorithm that achieves similar performances to that of [MMV'12]. Our algorithm runs in time $O(|V(G)|^{1+o(1)} + |E(G)|^{1+o(1)})$ and finds a balanced cut of value $O(α)$. Our approach appears easily extendible to related problem, such as Sparsest Cut, and also yields an near-linear time $O(1)$-approximation to Dagupta's objective function for hierarchical clustering [Dasgupta, STOC'16] for the semi-random hierarchical stochastic block model inputs of [Cohen-Addad, Kanade, Mallmann-Trenn, Mathieu, JACM'19].

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arXiv2024

Multi-View Stochastic Block Models

Graph clustering is a central topic in unsupervised learning with a multitude of practical applications. In recent years, multi-view graph clustering has gained a lot of attention for its applicability to real-world instances where one has access to multiple data sources. In this paper we formalize a new family of models, called \textit{multi-view stochastic block models} that captures this setting. For this model, we first study efficient algorithms that naively work on the union of multiple graphs. Then, we introduce a new efficient algorithm that provably outperforms previous approaches by analyzing the structure of each graph separately. Furthermore, we complement our results with an information-theoretic lower bound studying the limits of what can be done in this model. Finally, we corroborate our results with experimental evaluations.

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arXiv2024

Private graphon estimation via sum-of-squares

We develop the first pure node-differentially-private algorithms for learning stochastic block models and for graphon estimation with polynomial running time for any constant number of blocks. The statistical utility guarantees match those of the previous best information-theoretic (exponential-time) node-private mechanisms for these problems. The algorithm is based on an exponential mechanism for a score function defined in terms of a sum-of-squares relaxation whose level depends on the number of blocks. The key ingredients of our results are (1) a characterization of the distance between the block graphons in terms of a quadratic optimization over the polytope of doubly stochastic matrices, (2) a general sum-of-squares convergence result for polynomial optimization over arbitrary polytopes, and (3) a general approach to perform Lipschitz extensions of score functions as part of the sum-of-squares algorithmic paradigm.

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arXiv2024

Max-Cut with $ε$-Accurate Predictions

We study the approximability of the MaxCut problem in the presence of predictions. Specifically, we consider two models: in the noisy predictions model, for each vertex we are given its correct label in $\{-1,+1\}$ with some unknown probability $1/2 + ε$, and the other (incorrect) label otherwise. In the more-informative partial predictions model, for each vertex we are given its correct label with probability $ε$ and no label otherwise. We assume only pairwise independence between vertices in both models. We show how these predictions can be used to improve on the worst-case approximation ratios for this problem. Specifically, we give an algorithm that achieves an $α+ \widetildeΩ(ε^4)$-approximation for the noisy predictions model, where $α\approx 0.878$ is the MaxCut threshold. While this result also holds for the partial predictions model, we can also give a $β+ Ω(ε)$-approximation, where $β\approx 0.858$ is the approximation ratio for MaxBisection given by Raghavendra and Tan. This answers a question posed by Ola Svensson in his plenary session talk at SODA'23.

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arXiv2023

Private estimation algorithms for stochastic block models and mixture models

We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms. To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians. For the former, we present the first efficient $(ε, δ)$-differentially private algorithm for both weak recovery and exact recovery. Previously known algorithms achieving comparable guarantees required quasi-polynomial time. For the latter, we design an $(ε, δ)$-differentially private algorithm that recovers the centers of the $k$-mixture when the minimum separation is at least $ O(k^{1/t}\sqrt{t})$. For all choices of $t$, this algorithm requires sample complexity $n\geq k^{O(1)}d^{O(t)}$ and time complexity $(nd)^{O(t)}$. Prior work required minimum separation at least $O(\sqrt{k})$ as well as an explicit upper bound on the Euclidean norm of the centers.

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arXiv2023

Reaching Kesten-Stigum Threshold in the Stochastic Block Model under Node Corruptions

We study robust community detection in the context of node-corrupted stochastic block model, where an adversary can arbitrarily modify all the edges incident to a fraction of the $n$ vertices. We present the first polynomial-time algorithm that achieves weak recovery at the Kesten-Stigum threshold even in the presence of a small constant fraction of corrupted nodes. Prior to this work, even state-of-the-art robust algorithms were known to break under such node corruption adversaries, when close to the Kesten-Stigum threshold. We further extend our techniques to the $Z_2$ synchronization problem, where our algorithm reaches the optimal recovery threshold in the presence of similar strong adversarial perturbations. The key ingredient of our algorithm is a novel identifiability proof that leverages the push-out effect of the Grothendieck norm of principal submatrices.

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arXiv2023

A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs

We define a novel notion of ``non-backtracking'' matrix associated to any symmetric matrix, and we prove a ``Ihara-Bass'' type formula for it. We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with $k$ variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a $p$ fraction of assignments, if the instance contains $n$ variables and $n^{k/2} / ε^2$ constraints, we can efficiently compute a certificate that the optimum satisfies at most a $p+O_k(ε)$ fraction of constraints. Previously, this was known for even $k$, but for odd $k$ one needed $n^{k/2} (\log n)^{O(1)} / ε^2$ random constraints to achieve the same conclusion. Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck's inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is $o(n^{\lceil k/2 \rceil})$ and the third one cannot work when the number of constraints is $o(n^{k/2} \sqrt{\log n})$. We further apply our techniques to obtain a new PTAS finding assignments for $k$-CSP instances with $n^{k/2} / ε^2$ constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.

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arXiv2022

Higher degree sum-of-squares relaxations robust against oblivious outliers

We consider estimation models of the form $Y=X^*+N$, where $X^*$ is some $m$-dimensional signal we wish to recover, and $N$ is symmetrically distributed noise that may be unbounded in all but a small $α$ fraction of the entries. We introduce a family of algorithms that under mild assumptions recover the signal $X^*$ in all estimation problems for which there exists a sum-of-squares algorithm that succeeds in recovering the signal $X^*$ when the noise $N$ is Gaussian. This essentially shows that it is enough to design a sum-of-squares algorithm for an estimation problem with Gaussian noise in order to get the algorithm that works with the symmetric noise model. Our framework extends far beyond previous results on symmetric noise models and is even robust to adversarial perturbations. As concrete examples, we investigate two problems for which no efficient algorithms were known to work for heavy-tailed noise: tensor PCA and sparse PCA. For the former, our algorithm recovers the principal component in polynomial time when the signal-to-noise ratio is at least $\tilde{O}(n^{p/4}/α)$, that matches (up to logarithmic factors) current best known algorithmic guarantees for Gaussian noise. For the latter, our algorithm runs in quasipolynomial time and matches the state-of-the-art guarantees for quasipolynomial time algorithms in the case of Gaussian noise. Using a reduction from the planted clique problem, we provide evidence that the quasipolynomial time is likely to be necessary for sparse PCA with symmetric noise. In our proofs we use bounds on the covering numbers of sets of pseudo-expectations, which we obtain by certifying in sum-of-squares upper bounds on the Gaussian complexities of sets of solutions. This approach for bounding the covering numbers of sets of pseudo-expectations may be interesting in its own right and may find other application in future works.

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arXiv2022

Fast algorithm for overcomplete order-3 tensor decomposition

We develop the first fast spectral algorithm to decompose a random third-order tensor over $\mathbb{R}^d$ of rank up to $O(d^{3/2}/\text{polylog}(d))$. Our algorithm only involves simple linear algebra operations and can recover all components in time $O(d^{6.05})$ under the current matrix multiplication time. Prior to this work, comparable guarantees could only be achieved via sum-of-squares [Ma, Shi, Steurer 2016]. In contrast, fast algorithms [Hopkins, Schramm, Shi, Steurer 2016] could only decompose tensors of rank at most $O(d^{4/3}/\text{polylog}(d))$. Our algorithmic result rests on two key ingredients. A clean lifting of the third-order tensor to a sixth-order tensor, which can be expressed in the language of tensor networks. A careful decomposition of the tensor network into a sequence of rectangular matrix multiplications, which allows us to have a fast implementation of the algorithm.

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arXiv2022

On the well-spread property and its relation to linear regression

We consider the robust linear regression model $\boldsymbol{y} = Xβ^* + \boldsymbolη$, where an adversary oblivious to the design $X \in \mathbb{R}^{n \times d}$ may choose $\boldsymbolη$ to corrupt all but a (possibly vanishing) fraction of the observations $\boldsymbol{y}$ in an arbitrary way. Recent work [dLN+21, dNS21] has introduced efficient algorithms for consistent recovery of the parameter vector. These algorithms crucially rely on the design matrix being well-spread (a matrix is well-spread if its column span is far from any sparse vector). In this paper, we show that there exists a family of design matrices lacking well-spreadness such that consistent recovery of the parameter vector in the above robust linear regression model is information-theoretically impossible. We further investigate the average-case time complexity of certifying well-spreadness of random matrices. We show that it is possible to efficiently certify whether a given $n$-by-$d$ Gaussian matrix is well-spread if the number of observations is quadratic in the ambient dimension. We complement this result by showing rigorous evidence -- in the form of a lower bound against low-degree polynomials -- of the computational hardness of this same certification problem when the number of observations is $o(d^2)$.

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