Cybersecurity and Cyberforensics Conference最新文献

Finer-Grained Hardness of Kernel Density Estimation 更精细的核密度估计难度

Cybersecurity and Cyberforensics Conference Pub Date : 2024-07-02 DOI: 10.4230/LIPIcs.CCC.2024.35

Josh Alman, Yunfeng Guan

{"title":"Finer-Grained Hardness of Kernel Density Estimation","authors":"Josh Alman, Yunfeng Guan","doi":"10.4230/LIPIcs.CCC.2024.35","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2024.35","url":null,"abstract":"In batch Kernel Density Estimation (KDE) for a kernel function $f$, we are given as input $2n$ points $x^{(1)}, cdots, x^{(n)}, y^{(1)}, cdots, y^{(n)}$ in dimension $m$, as well as a vector $v in mathbb{R}^n$. These inputs implicitly define the $n times n$ kernel matrix $K$ given by $K[i,j] = f(x^{(i)}, y^{(j)})$. The goal is to compute a vector $v$ which approximates $K w$ with $|| Kw - v||_infty<varepsilon ||w||_1$. A recent line of work has proved fine-grained lower bounds conditioned on SETH. Backurs et al. first showed the hardness of KDE for Gaussian-like kernels with high dimension $m = Omega(log n)$ and large scale $B = Omega(log n)$. Alman et al. later developed new reductions in roughly this same parameter regime, leading to lower bounds for more general kernels, but only for very small error $varepsilon<2^{- log^{Omega(1)} (n)}$. In this paper, we refine the approach of Alman et al. to show new lower bounds in all parameter regimes, closing gaps between the known algorithms and lower bounds. In the setting where $m = Clog n$ and $B = o(log n)$, we prove Gaussian KDE requires $n^{2-o(1)}$ time to achieve additive error $varepsilon<Omega(m/B)^{-m}$, matching the performance of the polynomial method up to low-order terms. In the low dimensional setting $m = o(log n)$, we show that Gaussian KDE requires $n^{2-o(1)}$ time to achieve $varepsilon$ such that $log log (varepsilon^{-1})>tilde Omega ((log n)/m)$, matching the error bound achievable by FMM up to low-order terms. To our knowledge, no nontrivial lower bound was previously known in this regime. Our new lower bounds make use of an intricate analysis of a special case of the kernel matrix -- the `counting matrix'. As a key technical lemma, we give a novel approach to bounding the entries of its inverse by using Schur polynomials from algebraic combinatorics.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"26 1‐2","pages":"35:1-35:21"},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141686858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the algebraic proof complexity of Tensor Isomorphism 张量同构的代数证明复杂度

Cybersecurity and Cyberforensics Conference Pub Date : 2023-05-30 DOI: 10.48550/arXiv.2305.19320

Nicola Galesi, Joshua A. Grochow, T. Pitassi, Adrian She

{"title":"On the algebraic proof complexity of Tensor Isomorphism","authors":"Nicola Galesi, Joshua A. Grochow, T. Pitassi, Adrian She","doi":"10.48550/arXiv.2305.19320","DOIUrl":"https://doi.org/10.48550/arXiv.2305.19320","url":null,"abstract":"The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz&Grohe, SODA '17). Our main results are an $Omega(n)$ lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices, or deriving $BA=I$ from $AB=I$. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication $AB=I rightarrow BA=I$. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130777629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree 量子查询复杂度与多项式度的指数分离

Cybersecurity and Cyberforensics Conference Pub Date : 2023-01-22 DOI: 10.4230/LIPIcs.CCC.2023.24

A. Ambainis, Aleksandrs Belovs

引用次数: 3

The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate 变分量子算法的最优深度是qcma -难以近似

Cybersecurity and Cyberforensics Conference Pub Date : 2022-11-22 DOI: 10.4230/LIPIcs.CCC.2023.34

Lennart Bittel, Sevag Gharibian, M. Kliesch

引用次数: 4

A degree 4 sum-of-squares lower bound for the clique number of the Paley graph Paley图团数的4次平方和下界

Cybersecurity and Cyberforensics Conference Pub Date : 2022-11-04 DOI: 10.48550/arXiv.2211.02713

Dmitriy Kunisky, Xifan Yu

{"title":"A degree 4 sum-of-squares lower bound for the clique number of the Paley graph","authors":"Dmitriy Kunisky, Xifan Yu","doi":"10.48550/arXiv.2211.02713","DOIUrl":"https://doi.org/10.48550/arXiv.2211.02713","url":null,"abstract":"We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number $p$ of vertices has value at least $Omega(p^{1/3})$. This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is $O(mathrm{polylog}(p))$. Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to $mathrm{polylog}(p)$ terms) with high probability for the ErdH{o}s-R'{e}nyi random graph on $p$ vertices, whose clique number is with high probability $O(log(p))$. We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as $O(p^{1/2 - epsilon})$ for some $epsilon>0$, and give a matrix norm calculation indicating that the pseudocalibration proof strategy for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the\"$sqrt{p}$ barrier\"for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from $1/2$ to $1/3$.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121600485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

A Distribution Testing Oracle Separating QMA and QCMA 一个分离QMA和QCMA的分布测试Oracle

Cybersecurity and Cyberforensics Conference Pub Date : 2022-10-27 DOI: 10.4230/LIPIcs.CCC.2023.22

A. Natarajan, Chinmay Nirkhe

引用次数: 1

Translationally Invariant Constraint Optimization Problems 平移不变约束优化问题

Cybersecurity and Cyberforensics Conference Pub Date : 2022-09-19 DOI: 10.48550/arXiv.2209.08731

D. Aharonov, S. Irani

{"title":"Translationally Invariant Constraint Optimization Problems","authors":"D. Aharonov, S. Irani","doi":"10.48550/arXiv.2209.08731","DOIUrl":"https://doi.org/10.48550/arXiv.2209.08731","url":null,"abstract":"We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the complexity of function versions of such problems, with the additional restriction that the constraints are translationally invariant, namely, the variables are located at the vertices of a 2D grid and the constraint between every pair of adjacent variables is the same in each dimension. The only input to the problem is thus the size of the grid. This problem is equivalent to one of the most interesting problems in classical physics, namely, computing the lowest energy of a classical system of particles on the grid. We provide a tight characterization of the complexity of this problem, and show that it is complete for the class $FP^{NEXP}$. Gottesman and Irani (FOCS 2009) also studied classical translationally-invariant constraint satisfaction problems; they show that the problem of deciding whether the cost of the optimal solution is below a given threshold is NEXP-complete. Our result is thus a strengthening of their result from the decision version to the function version of the problem. Our result can also be viewed as a generalization to the translationally invariant setting, of Krentel's famous result from 1988, showing that the function version of SAT is complete for the class $FP^{NP}$. An essential ingredient in the proof is a study of the complexity of a gapped variant of the problem. We show that it is NEXP-hard to approximate the cost of the optimal assignment to within an additive error of $Omega(N^{1/4})$, for an $N times N$ grid. To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. As a byproduct of our results, we also show that a decision version of the optimization problem which asks whether the cost of the optimal assignment is odd or even is also complete for $P^{NEXP}$.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"31 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133072015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors 汉明和插入-删除错误的松弛局部可译码

Cybersecurity and Cyberforensics Conference Pub Date : 2022-09-19 DOI: 10.48550/arXiv.2209.08688

Alexander R. Block, Jeremiah Blocki, Kuan Cheng, Elena Grigorescu, Xin Li, Yu Zheng, Minshen Zhu

{"title":"On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors","authors":"Alexander R. Block, Jeremiah Blocki, Kuan Cheng, Elena Grigorescu, Xin Li, Yu Zheng, Minshen Zhu","doi":"10.48550/arXiv.2209.08688","DOIUrl":"https://doi.org/10.48550/arXiv.2209.08688","url":null,"abstract":"Locally Decodable Codes (LDCs) are error-correcting codes $C:Sigma^nrightarrow Sigma^m$ with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson et al. showed how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Insdel LDCs were first studied by Ostrovsky and Paskin-Cherniavsky, and are further motivated by recent advances in DNA random access bio-technologies, in which the goal is to retrieve individual files from a DNA storage database. Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries, over the binary alphabet. This answers a question explicitly raised by Gur and Lachish. Our result exhibits a\"phase-transition\"-type behavior on the codeword length for constant-query Hamming RLDCs. We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with with parameters matching those of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133177797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

An Improved Trickle-Down Theorem for Partite Complexes 局部配合物的一个改进的滴下定理

Cybersecurity and Cyberforensics Conference Pub Date : 2022-08-09 DOI: 10.48550/arXiv.2208.04486

Dorna Abdolazimi, S. Gharan

引用次数: 1

Hardness of Function Composition for Semantic Read once Branching Programs 语义读取一次分支程序的函数合成难易度

Cybersecurity and Cyberforensics Conference Pub Date : 2018-06-22 DOI: 10.4230/LIPIcs.CCC.2018.15

J. Edmonds, Venkatesh Medabalimi, T. Pitassi

引用次数: 2