4 Search Results for "Max, Nelson"

Document
DOI: 10.4230/LIPIcs.ITCS.2023.39
Generalized Private Selection and Testing with High Confidence

Authors: Edith Cohen, Xin Lyu, Jelani Nelson, Tamás Sarlós, and Uri Stemmer

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
Composition theorems are general and powerful tools that facilitate privacy accounting across multiple data accesses from per-access privacy bounds. However they often result in weaker bounds compared with end-to-end analysis. Two popular tools that mitigate that are the exponential mechanism (or report noisy max) and the sparse vector technique, generalized in a recent private selection framework by Liu and Talwar (STOC 2019). In this work, we propose a flexible framework of private selection and testing that generalizes the one proposed by Liu and Talwar, supporting a wide range of applications. We apply our framework to solve several fundamental tasks, including query releasing, top-k selection, and stable selection, with improved confidence-accuracy tradeoffs. Additionally, for online settings, we apply our private testing to design a mechanism for adaptive query releasing, which improves the sample complexity dependence on the confidence parameter for the celebrated private multiplicative weights algorithm of Hardt and Rothblum (FOCS 2010).
Cite as

Edith Cohen, Xin Lyu, Jelani Nelson, Tamás Sarlós, and Uri Stemmer. Generalized Private Selection and Testing with High Confidence. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cohen_et_al:LIPIcs.ITCS.2023.39,
  author =	{Cohen, Edith and Lyu, Xin and Nelson, Jelani and Sarl\'{o}s, Tam\'{a}s and Stemmer, Uri},
  title =	{{Generalized Private Selection and Testing with High Confidence}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{39:1--39:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.39},
  URN =		{urn:nbn:de:0030-drops-175426},
  doi =		{10.4230/LIPIcs.ITCS.2023.39},
  annote =	{Keywords: differential privacy, sparse vector technique, adaptive data analysis}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.77
Deterministic Sparse Fourier Transform with an 𝓁_{∞} Guarantee

Authors: Yi Li and Vasileios Nakos

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of x ∈ ℂⁿ and design a recovery algorithm such that the output of the algorithm approximates x̂, the Discrete Fourier Transform (DFT) of x. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al. (J Fourier Anal Appl 2018), which obtains O(k² log^(-1) k ⋅ log^5.5 n) samples and a similar runtime with the 𝓁₂/𝓁₁ guarantee. We focus on the stronger 𝓁_∞/𝓁₁ guarantee and the closely related problem of incoherent matrices. We list our contributions as follows. 1) We find a deterministic collection of O(k² log n) samples for the 𝓁_∞/𝓁₁ recovery in time O(nk log² n), and a deterministic collection of O(k² log² n) samples for the 𝓁_∞/𝓁₁ sparse recovery in time O(k² log³n). 2) We give new deterministic constructions of incoherent matrices that are row-sampled submatrices of the DFT matrix, via a derandomization of Bernstein’s inequality and bounds on exponential sums considered in analytic number theory. Our first construction matches a previous randomized construction of Nelson, Nguyen and Woodruff (RANDOM'12), where there was no constraint on the form of the incoherent matrix. Our algorithms are nearly sample-optimal, since a lower bound of Ω(k² + k log n) is known, even for the case where the sensing matrix can be arbitrarily designed. A similar lower bound of Ω(k² log n/ log k) is known for incoherent matrices.
Cite as

Yi Li and Vasileios Nakos. Deterministic Sparse Fourier Transform with an 𝓁_{∞} Guarantee. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{li_et_al:LIPIcs.ICALP.2020.77,
  author =	{Li, Yi and Nakos, Vasileios},
  title =	{{Deterministic Sparse Fourier Transform with an 𝓁\underline\{∞\} Guarantee}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{77:1--77:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.77},
  URN =		{urn:nbn:de:0030-drops-124844},
  doi =		{10.4230/LIPIcs.ICALP.2020.77},
  annote =	{Keywords: Fourier sparse recovery, derandomization, incoherent matrices}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.44
An Improved Analysis of the ER-SpUD Dictionary Learning Algorithm

Authors: Jaroslaw Blasiok and Jelani Nelson

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
In dictionary learning we observe Y = AX + E for some Y in R^{n*p}, A in R^{m*n}, and X in R^{m*p}, where p >= max{n, m}, and typically m >=n. The matrix Y is observed, and A, X, E are unknown. Here E is a "noise" matrix of small norm, and X is column-wise sparse. The matrix A is referred to as a dictionary, and its columns as atoms. Then, given some small number p of samples, i.e. columns of Y , the goal is to learn the dictionary A up to small error, as well as the coefficient matrix X. In applications one could for example think of each column of Y as a distinct image in a database. The motivation is that in many applications data is expected to sparse when represented by atoms in the "right" dictionary A (e.g. images in the Haar wavelet basis), and the goal is to learn A from the data to then use it for other applications. Recently, the work of [Spielman/Wang/Wright, COLT'12] proposed the dictionary learning algorithm ER-SpUD with provable guarantees when E = 0 and m = n. That work showed that if X has independent entries with an expected Theta n non-zeroes per column for 1/n <~ Theta <~ 1/sqrt(n), and with non-zero entries being subgaussian, then for p >~ n^2 log^2 n with high probability ER-SpUD outputs matrices A', X' which equal A, X up to permuting and scaling columns (resp. rows) of A (resp. X). They conjectured that p >~ n log n suffices, which they showed was information theoretically necessary for any algorithm to succeed when Theta =~ 1/n. Significant progress toward showing that p >~ n log^4 n might suffice was later obtained in [Luh/Vu, FOCS'15]. In this work, we show that for a slight variant of ER-SpUD, p >~ n log(n/delta) samples suffice for successful recovery with probability 1 - delta. We also show that without our slight variation made to ER-SpUD, p >~ n^{1.99} samples are required even to learn A, X with a small success probability of 1/ poly(n). This resolves the main conjecture of [Spielman/Wang/Wright, COLT'12], and contradicts a result of [Luh/Vu, FOCS'15], which claimed that p >~ n log^4 n guarantees high probability of success for the original ER-SpUD algorithm.
Cite as

Jaroslaw Blasiok and Jelani Nelson. An Improved Analysis of the ER-SpUD Dictionary Learning Algorithm. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{blasiok_et_al:LIPIcs.ICALP.2016.44,
  author =	{Blasiok, Jaroslaw and Nelson, Jelani},
  title =	{{An Improved Analysis of the ER-SpUD Dictionary Learning Algorithm}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{44:1--44:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.44},
  URN =		{urn:nbn:de:0030-drops-63246},
  doi =		{10.4230/LIPIcs.ICALP.2016.44},
  annote =	{Keywords: dictionary learning, stochastic processes, generic chaining}
}
Document
DOI: 10.4230/DFU.SciViz.2010.259
Local and Global Illumination in the Volume Rendering Integral

Authors: Nelson Max and Min Chen

Published in: Dagstuhl Follow-Ups, Volume 1, Scientific Visualization: Advanced Concepts (2010)

Abstract
This article is intended as an update of the major survey by Max [Max, "Optical models for direct volume rendering.", IEEE Trans. on Visualization and Computer Graphics, 1(2):99–108, 1995] on optical models for direct volume rendering. It provides a brief overview of the subject scope covered by [Max, "Optical models for direct volume rendering.", IEEE Trans. on Visualization and Computer Graphics, 1(2):99–108, 1995], and brings recent developments, such as new shadow algorithms and refraction rendering, into the perspective. In particular, we examine three fundamentals aspects of direct volume rendering, namely the volume rendering integral, local illumination models and global illumination models, in a wavelength-independent manner. We review the developments on spectral volume rendering, in which visible light are considered as a form of electromagnetic radiation, optical models are implemented in conjunction with representations of spectral power distribution. This survey can provide a basis for, and encourage, new efforts for developing and using complex illumination models to achieve better realism and perception through optical correctness.
Cite as

Nelson Max and Min Chen. Local and Global Illumination in the Volume Rendering Integral. In Scientific Visualization: Advanced Concepts. Dagstuhl Follow-Ups, Volume 1, pp. 259-274, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InCollection{max_et_al:DFU.SciViz.2010.259,
  author =	{Max, Nelson and Chen, Min},
  title =	{{Local and Global Illumination in the Volume Rendering Integral}},
  booktitle =	{Scientific Visualization: Advanced Concepts},
  pages =	{259--274},
  series =	{Dagstuhl Follow-Ups},
  ISBN =	{978-3-939897-19-4},
  ISSN =	{1868-8977},
  year =	{2010},
  volume =	{1},
  editor =	{Hagen, Hans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DFU.SciViz.2010.259},
  URN =		{urn:nbn:de:0030-drops-27090},
  doi =		{10.4230/DFU.SciViz.2010.259},
  annote =	{Keywords: Volume Rendering, Illumination Model}
}
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