3 Search Results for "Aggarwal, Amol"

Document
DOI: 10.4230/LIPIcs.ITCS.2025.14
The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions

Authors: Huck Bennett, Surendra Ghentiyala, and Noah Stephens-Davidowitz

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We show a number of connections between two types of search problems: (1) the problem of finding an L-wise multicollision in the output of a function; and (2) the problem of finding two codewords in a code (or two vectors in a lattice) that are within distance d of each other. Specifically, we study these problems in the total regime, in which L and d are chosen so that such a solution is guaranteed to exist, though it might be hard to find. In more detail, we study the total search problem in which the input is a function 𝒞 : [A] → [B] (represented as a circuit) and the goal is to find L ≤ ⌈A/B⌉ distinct elements x_1,…, x_L ∈ A such that 𝒞(x_1) = ⋯ = 𝒞(x_L). The associated complexity classes Polynomial Multi-Pigeonhole Principle ((A,B)-PMPP^L) consist of all problems that reduce to this problem. We show close connections between (A,B)-PMPP^L and many celebrated upper bounds on the minimum distance of a code or lattice (and on the list-decoding radius). In particular, we show that the associated computational problems (i.e., the problem of finding two distinct codewords or lattice points that are close to each other) are in (A,B)-PMPP^L, with a more-or-less smooth tradeoff between the distance d and the parameters A, B, and L. These connections are particularly rich in the case of codes, in which case we show that multiple incomparable bounds on the minimum distance lie in seemingly incomparable complexity classes. Surprisingly, we also show that the computational problems associated with some bounds on the minimum distance of codes are actually hard for these classes (for codes represented by arbitrary circuits). In fact, we show that finding two vectors within a certain distance d is actually hard for the important (and well-studied) class PWPP = (B²,B)-PMPP² in essentially all parameter regimes for which an efficient algorithm is not known, so that our hardness results are essentially tight. In fact, for some d (depending on the block length, message length, and alphabet size), we obtain both hardness and containment. We therefore completely settle the complexity of this problem for such parameters and add coding problems to the short list of problems known to be complete for PWPP. We also study (A,B)-PMPP^L as an interesting family of complexity classes in its own right, and we uncover a rich structure. Specifically, we use recent techniques from the cryptographic literature on multicollision-resistant hash functions to (1) show inclusions of the form (A,B)-PMPP^L ⊆ (A',B')-PMPP^L' for certain non-trivial parameters; (2) black-box separations between such classes in different parameter regimes; and (3) a non-black-box proof that (A,B)-PMPP^L ∈ FP if (A',B')-PMPP^L' ∈ FP for yet another parameter regime. We also show that (A,B)-PMPP^L lies in the recently introduced complexity class Polynomial Long Choice for some parameters.
Cite as

Huck Bennett, Surendra Ghentiyala, and Noah Stephens-Davidowitz. The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bennett_et_al:LIPIcs.ITCS.2025.14,
  author =	{Bennett, Huck and Ghentiyala, Surendra and Stephens-Davidowitz, Noah},
  title =	{{The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{14:1--14:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.14},
  URN =		{urn:nbn:de:0030-drops-226424},
  doi =		{10.4230/LIPIcs.ITCS.2025.14},
  annote =	{Keywords: Multicollisions, Error-correcting codes, Lattices}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.35
Finer-Grained Hardness of Kernel Density Estimation

Authors: Josh Alman and Yunfeng Guan

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
In batch Kernel Density Estimation (KDE) for a kernel function f : ℝ^m × ℝ^m → ℝ, we are given as input 2n points x^{(1)}, …, x^{(n)}, y^{(1)}, …, y^{(n)} ∈ ℝ^m in dimension m, as well as a vector v ∈ ℝⁿ. These inputs implicitly define the n × n kernel matrix K given by K[i,j] = f(x^{(i)}, y^{(j)}). The goal is to compute a vector v ∈ ℝⁿ which approximates K w, i.e., with || Kw - v||_∞ < ε ||w||₁. For illustrative purposes, consider the Gaussian kernel f(x,y) : = e^{-||x-y||₂²}. The classic approach to this problem is the famous Fast Multipole Method (FMM), which runs in time n ⋅ O(log^m(ε^{-1})) and is particularly effective in low dimensions because of its exponential dependence on m. Recently, as the higher-dimensional case m ≥ Ω(log n) has seen more applications in machine learning and statistics, new algorithms have focused on this setting: an algorithm using discrepancy theory, which runs in time O(n / ε), and an algorithm based on the polynomial method, which achieves inverse polynomial accuracy in almost linear time when the input points have bounded square diameter B < o(log n). A recent line of work has proved fine-grained lower bounds, with the goal of showing that the "curse of dimensionality" arising in FMM is necessary assuming the Strong Exponential Time Hypothesis (SETH). Backurs et al. [NeurIPS 2017] first showed the hardness of a variety of Empirical Risk Minimization problems including KDE for Gaussian-like kernels in the case with high dimension m = Ω(log n) and large scale B = Ω(log n). Alman et al. [FOCS 2020] later developed new reductions in roughly this same parameter regime, leading to lower bounds for more general kernels, but only for very small error ε < 2^{- log^{Ω(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. For example: - 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 ε < Ω(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 ε such that log log (ε^{-1}) > ̃ Ω ((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 approach also generalizes to any parameter regime and any kernel. For example, we achieve similar fine-grained hardness results for any kernel with slowly-decaying Taylor coefficients such as the Cauchy kernel. Our new lower bounds make use of an intricate analysis of the "counting matrix", a special case of the kernel matrix focused on carefully-chosen evaluation points. As a key technical lemma, we give a novel approach to bounding the entries of its inverse by using Schur polynomials from algebraic combinatorics.
Cite as

Josh Alman and Yunfeng Guan. Finer-Grained Hardness of Kernel Density Estimation. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 35:1-35:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alman_et_al:LIPIcs.CCC.2024.35,
  author =	{Alman, Josh and Guan, Yunfeng},
  title =	{{Finer-Grained Hardness of Kernel Density Estimation}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{35:1--35:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.35},
  URN =		{urn:nbn:de:0030-drops-204311},
  doi =		{10.4230/LIPIcs.CCC.2024.35},
  annote =	{Keywords: Kernel Density Estimation, Fine-Grained Complexity, Schur Polynomials}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.22
Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation

Authors: Amol Aggarwal and Josh Alman

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
For any real numbers B ≥ 1 and δ ∈ (0,1) and function f: [0,B] → ℝ, let d_{B; δ}(f) ∈ ℤ_{> 0} denote the minimum degree of a polynomial p(x) satisfying sup_{x ∈ [0,B]} |p(x) - f(x)| < δ. In this paper, we provide precise asymptotics for d_{B; δ}(e^{-x}) and d_{B; δ}(e^x) in terms of both B and δ, improving both the previously known upper bounds and lower bounds. In particular, we show d_{B; δ}(e^{-x}) = Θ(max{√{B log(δ^{-1})}, log(δ^{-1})/{log(B^{-1} log(δ^{-1}))}}), and d_{B; δ}(e^{x}) = Θ(max{B, log(δ^{-1})/{log(B^{-1} log(δ^{-1}))}}), and we explicitly determine the leading coefficients in most parameter regimes. Polynomial approximations for e^{-x} and e^x have applications to the design of algorithms for many problems, including in scientific computing, graph algorithms, machine learning, and statistics. Our degree bounds show both the power and limitations of these algorithms. We focus in particular on the Batch Gaussian Kernel Density Estimation problem for n sample points in Θ(log n) dimensions with error δ = n^{-Θ(1)}. We show that the running time one can achieve depends on the square of the diameter of the point set, B, with a transition at B = Θ(log n) mirroring the corresponding transition in d_{B; δ}(e^{-x}): - When B = o(log n), we give the first algorithm running in time n^{1 + o(1)}. - When B = κ log n for a small constant κ > 0, we give an algorithm running in time n^{1 + O(log log κ^{-1} /log κ^{-1})}. The log log κ^{-1} /log κ^{-1} term in the exponent comes from analyzing the behavior of the leading constant in our computation of d_{B; δ}(e^{-x}). - When B = ω(log n), we show that time n^{2 - o(1)} is necessary assuming SETH.
Cite as

Amol Aggarwal and Josh Alman. Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aggarwal_et_al:LIPIcs.CCC.2022.22,
  author =	{Aggarwal, Amol and Alman, Josh},
  title =	{{Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{22:1--22:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.22},
  URN =		{urn:nbn:de:0030-drops-165846},
  doi =		{10.4230/LIPIcs.CCC.2022.22},
  annote =	{Keywords: polynomial approximation, kernel density estimation, Chebyshev polynomials}
}
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