17 Search Results for "Chattopadhyay, Eshan"

Document
DOI: 10.4230/LIPIcs.ITCS.2024.28
Extractors for Polynomial Sources over 𝔽₂

Authors: Eshan Chattopadhyay, Jesse Goodman, and Mohit Gurumukhani

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We explicitly construct the first nontrivial extractors for degree d ≥ 2 polynomial sources over 𝔽₂. Our extractor requires min-entropy k ≥ n - (√{log n})/((log log n / d)^{d/2}). Previously, no constructions were known, even for min-entropy k ≥ n-1. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy k can be generated by O(k) uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below k ≥ n-o(n). In more detail, we show that sumset extractors cannot even disperse from degree 2 polynomial sources with min-entropy k ≥ n-O(n/log log n). In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.
Cite as

Eshan Chattopadhyay, Jesse Goodman, and Mohit Gurumukhani. Extractors for Polynomial Sources over 𝔽₂. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2024.28,
  author =	{Chattopadhyay, Eshan and Goodman, Jesse and Gurumukhani, Mohit},
  title =	{{Extractors for Polynomial Sources over \mathbb{F}₂}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{28:1--28:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.28},
  URN =		{urn:nbn:de:0030-drops-195569},
  doi =		{10.4230/LIPIcs.ITCS.2024.28},
  annote =	{Keywords: Extractors, low-degree polynomials, varieties, sumset extractors}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.29
Recursive Error Reduction for Regular Branching Programs

Authors: Eshan Chattopadhyay and Jyun-Jie Liao

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved error reduction framework for the derandomization of regular read-once branching programs (ROBPs). Their result is based on a clever modification to the inverse Laplacian perspective of space-bounded derandomization, which was originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and Vadhan (FOCS 2020). In this work, we give an alternative error reduction framework for regular ROBPs. Our new framework is based on a binary recursive formula from the work of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted pseudorandom generators (WPRGs) for general ROBPs. Based on our new error reduction framework, we give alternative proofs to the following results for regular ROBPs of length n and width w, both of which were proved in the work of Chen et al. using their error reduction: - There is a WPRG with error ε that has seed length Õ(log(n)(√{log(1/ε)}+log(w))+log(1/ε)). - There is a (non-black-box) deterministic algorithm which estimates the expectation of any such program within error ±ε with space complexity Õ(log(nw)⋅log log(1/ε)). This was first proved in the work of Ahmadinejad et al., but the proof by Chen et al. is simpler. Because of the binary recursive nature of our new framework, both of our proofs are based on a straightforward induction that is arguably simpler than the Laplacian-based proof in the work of Chen et al. In fact, because of its simplicity, our proof of the second result directly gives a slightly stronger claim: our algorithm computes a ε-singular value approximation (a notion of approximation introduced in a recent work by Ahmadinejad, Peebles, Pyne, Sidford and Vadhan (FOCS 2023)) of the random walk matrix of the given ROBP in space Õ(log(nw)⋅log log(1/ε)). It is not clear how to get this stronger result from the previous proofs.
Cite as

Eshan Chattopadhyay and Jyun-Jie Liao. Recursive Error Reduction for Regular Branching Programs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2024.29,
  author =	{Chattopadhyay, Eshan and Liao, Jyun-Jie},
  title =	{{Recursive Error Reduction for Regular Branching Programs}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{29:1--29:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.29},
  URN =		{urn:nbn:de:0030-drops-195571},
  doi =		{10.4230/LIPIcs.ITCS.2024.29},
  annote =	{Keywords: read-once branching program, regular branching program, weighted pseudorandom generator, derandomization}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.9
Hardness Against Linear Branching Programs and More

Authors: Eshan Chattopadhyay and Jyun-Jie Liao

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
In a recent work, Gryaznov, Pudlák and Talebanfard (CCC '22) introduced a linear variant of read-once branching programs, with motivations from circuit and proof complexity. Such a read-once linear branching program is a branching program where each node is allowed to make 𝔽₂-linear queries, and is read-once in the sense that the queries on each path is linearly independent. As their main result, they constructed an explicit function with average-case complexity 2^{n/3-o(n)} against a slightly restricted model, which they call strongly read-once linear branching programs. The main tool in their lower bound result is a new type of extractor, called directional affine extractors, that they introduced. Our main result is an explicit function with 2^{n-o(n)} average-case complexity against the strongly read-once linear branching program model, which is almost optimal. This result is based on a new connection from this problem to sumset extractors, which is a randomness extractor model introduced by Chattopadhyay and Li (STOC '16) as a generalization of many other well-studied models including two-source extractors, affine extractors and small-space extractors. With this new connection, our lower bound naturally follows from a recent construction of sumset extractors by Chattopadhyay and Liao (STOC '22). In addition, we show that directional affine extractors imply sumset extractors in a restricted setting. We observe that such restricted sumset sources are enough to derive lower bounds, and obtain an arguably more modular proof of the lower bound by Gryaznov, Pudlák and Talebanfard. We also initiate a study of pseudorandomness against linear branching programs. Our main result here is a hitting set generator construction against regular linear branching programs with constant width. We derive this result based on a connection to Kakeya sets over finite fields.
Cite as

Eshan Chattopadhyay and Jyun-Jie Liao. Hardness Against Linear Branching Programs and More. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 9:1-9:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2023.9,
  author =	{Chattopadhyay, Eshan and Liao, Jyun-Jie},
  title =	{{Hardness Against Linear Branching Programs and More}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{9:1--9:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.9},
  URN =		{urn:nbn:de:0030-drops-182794},
  doi =		{10.4230/LIPIcs.CCC.2023.9},
  annote =	{Keywords: linear branching programs, circuit lower bound, sumset extractors, hitting sets}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.10
Low-Degree Polynomials Extract From Local Sources

Authors: Omar Alrabiah, Eshan Chattopadhyay, Jesse Goodman, Xin Li, and João Ribeiro

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, this model was introduced by De and Watson (TOCT 2012) and Viola (SICOMP 2014), and is closely related to sources generated by AC⁰ circuits and bounded-width branching programs. In particular, extractors for local sources also work for sources generated by these classical computational models. Despite being introduced a decade ago, little progress has been made on improving the entropy requirement for extracting from local sources. The current best explicit extractors require entropy n^{1/2}, and follow via a reduction to affine extractors. To start, we prove a barrier showing that one cannot hope to improve this entropy requirement via a black-box reduction of this form. In particular, new techniques are needed. In our main result, we seek to answer whether low-degree polynomials (over 𝔽₂) hold potential for breaking this barrier. We answer this question in the positive, and fully characterize the power of low-degree polynomials as extractors for local sources. More precisely, we show that a random degree r polynomial is a low-error extractor for n-bit local sources with min-entropy Ω(r(nlog n)^{1/r}), and we show that this is tight. Our result leverages several new ingredients, which may be of independent interest. Our existential result relies on a new reduction from local sources to a more structured family, known as local non-oblivious bit-fixing sources. To show its tightness, we prove a "local version" of a structural result by Cohen and Tal (RANDOM 2015), which relies on a new "low-weight" Chevalley-Warning theorem.
Cite as

Omar Alrabiah, Eshan Chattopadhyay, Jesse Goodman, Xin Li, and João Ribeiro. Low-Degree Polynomials Extract From Local Sources. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{alrabiah_et_al:LIPIcs.ICALP.2022.10,
  author =	{Alrabiah, Omar and Chattopadhyay, Eshan and Goodman, Jesse and Li, Xin and Ribeiro, Jo\~{a}o},
  title =	{{Low-Degree Polynomials Extract From Local Sources}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.10},
  URN =		{urn:nbn:de:0030-drops-163519},
  doi =		{10.4230/LIPIcs.ICALP.2022.10},
  annote =	{Keywords: Randomness extractors, local sources, samplable sources, AC⁰ circuits, branching programs, low-degree polynomials, Chevalley-Warning}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.40
The Space Complexity of Sampling

Authors: Eshan Chattopadhyay, Jesse Goodman, and David Zuckerman

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
Recently, there has been exciting progress in understanding the complexity of distributions. Here, the goal is to quantify the resources required to generate (or sample) a distribution. Proving lower bounds in this new setting is more challenging than in the classical setting, and has yielded interesting new techniques and surprising applications. In this work, we initiate a study of the complexity of sampling with limited memory, and obtain the first nontrivial sampling lower bounds against oblivious read-once branching programs (ROBPs). In our first main result, we show that any distribution sampled by an ROBP of width 2^{Ω(n)} has statistical distance 1-2^{-Ω(n)} from any distribution that is uniform over a good code. More generally, we obtain sampling lower bounds for any list decodable code, which are nearly tight. Previously, such a result was only known for sampling in AC⁰ (Lovett and Viola, CCC'11; Beck, Impagliazzo and Lovett, FOCS'12). As an application of our result, a known connection implies new data structure lower bounds for storing codewords. In our second main result, we prove a direct product theorem for sampling with ROBPs. Previously, no direct product theorems were known for the task of sampling, for any computational model. A key ingredient in our proof is a simple new lemma about amplifying statistical distance between sequences of somewhat-dependent random variables. Using this lemma, we also obtain a simple new proof of a known lower bound for sampling disjoint sets using two-party communication protocols (Göös and Watson, RANDOM'19).
Cite as

Eshan Chattopadhyay, Jesse Goodman, and David Zuckerman. The Space Complexity of Sampling. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 40:1-40:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2022.40,
  author =	{Chattopadhyay, Eshan and Goodman, Jesse and Zuckerman, David},
  title =	{{The Space Complexity of Sampling}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{40:1--40:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.40},
  URN =		{urn:nbn:de:0030-drops-156366},
  doi =		{10.4230/LIPIcs.ITCS.2022.40},
  annote =	{Keywords: Complexity of distributions, complexity of sampling, extractors, list decodable codes, lower bounds, read-once branching programs, small-space computation}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.52
Lower Bounds for XOR of Forrelations

Authors: Uma Girish, Ran Raz, and Wei Zhan

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models. To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations: Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021]. Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
Cite as

Uma Girish, Ran Raz, and Wei Zhan. Lower Bounds for XOR of Forrelations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{girish_et_al:LIPIcs.APPROX/RANDOM.2021.52,
  author =	{Girish, Uma and Raz, Ran and Zhan, Wei},
  title =	{{Lower Bounds for XOR of Forrelations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.52},
  URN =		{urn:nbn:de:0030-drops-147453},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.52},
  annote =	{Keywords: Forrelation, Quasipolynomial, Separation, Quantum versus Classical, Xor}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.53
Fourier Growth of Structured 𝔽₂-Polynomials and Applications

Authors: Jarosław Błasiok, Peter Ivanov, Yaonan Jin, Chin Ho Lee, Rocco A. Servedio, and Emanuele Viola

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We analyze the Fourier growth, i.e. the L₁ Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F₂-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013]. - We show that any read-Δ degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ (k Δ d)^{O(k)}. - We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F₂-polynomials.
Cite as

Jarosław Błasiok, Peter Ivanov, Yaonan Jin, Chin Ho Lee, Rocco A. Servedio, and Emanuele Viola. Fourier Growth of Structured 𝔽₂-Polynomials and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 53:1-53:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blasiok_et_al:LIPIcs.APPROX/RANDOM.2021.53,
  author =	{B{\l}asiok, Jaros{\l}aw and Ivanov, Peter and Jin, Yaonan and Lee, Chin Ho and Servedio, Rocco A. and Viola, Emanuele},
  title =	{{Fourier Growth of Structured \mathbb{F}₂-Polynomials and Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{53:1--53:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.53},
  URN =		{urn:nbn:de:0030-drops-147462},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.53},
  annote =	{Keywords: Fourier analysis, Pseudorandomness, Fourier growth}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.10
Fractional Pseudorandom Generators from Any Fourier Level

Authors: Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, and Abhishek Shetty

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al. [Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2019] that exploit L₁ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k. This interpolates previous works, which either require Fourier bounds on all levels [Chattopadhyay et al., 2019], or have polynomial dependence on the error parameter in the seed length [Eshan Chattopadhyay et al., 2019], and thus answers an open question in [Eshan Chattopadhyay et al., 2019]. As an example, we show that for polynomial error, Fourier bounds on the first O(log n) levels is sufficient to recover the seed length in [Chattopadhyay et al., 2019], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor’s theorem and bounding the degree-k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the level-k unsigned Fourier sum, which is potentially a much smaller quantity than the L₁ notion in previous works. By generalizing a connection established in [Chattopadhyay et al., 2020], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for 𝔽₂ polynomials with seed length close to the state-of-the-art construction due to Viola [Emanuele Viola, 2009].
Cite as

Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, and Abhishek Shetty. Fractional Pseudorandom Generators from Any Fourier Level. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2021.10,
  author =	{Chattopadhyay, Eshan and Gaitonde, Jason and Lee, Chin Ho and Lovett, Shachar and Shetty, Abhishek},
  title =	{{Fractional Pseudorandom Generators from Any Fourier Level}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.10},
  URN =		{urn:nbn:de:0030-drops-142843},
  doi =		{10.4230/LIPIcs.CCC.2021.10},
  annote =	{Keywords: Derandomization, pseudorandomness, pseudorandom generators, Fourier analysis}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.12
Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing

Authors: Oded Goldreich and Avi Wigderson

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
A graph G is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G = (V,E) is robustly self-ordered if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation π:V → V is proportional to the number of non-fixed-points of π. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph is expanding. The second construction is iterative, boosting the property of robust self-ordering from smaller to larger graphs. Structuraly, the first construction always yields expanding graphs, while the second construction may produce graphs that have many tiny (sub-logarithmic) connected components. We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors (with very weak parameters but with some additional natural features). We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.
Cite as

Oded Goldreich and Avi Wigderson. Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 12:1-12:74, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goldreich_et_al:LIPIcs.CCC.2021.12,
  author =	{Goldreich, Oded and Wigderson, Avi},
  title =	{{Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{12:1--12:74},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.12},
  URN =		{urn:nbn:de:0030-drops-142867},
  doi =		{10.4230/LIPIcs.CCC.2021.12},
  annote =	{Keywords: Asymmetric graphs, expanders, testing graph properties, two-source extractors, non-malleable extractors, coding theory, tolerant testing, random graphs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.22
Error Reduction for Weighted PRGs Against Read Once Branching Programs

Authors: Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, and Amnon Ta-Shma

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
Weighted pseudorandom generators (WPRGs), introduced by Braverman, Cohen and Garg [Braverman et al., 2020], are a generalization of pseudorandom generators (PRGs) in which arbitrary real weights are considered, rather than a probability mass. Braverman et al. constructed WPRGs against read once branching programs (ROBPs) with near-optimal dependence on the error parameter. Chattopadhyay and Liao [Eshan Chattopadhyay and Jyun-Jie Liao, 2020] somewhat simplified the technically involved BCG construction, also obtaining some improvement in parameters. In this work we devise an error reduction procedure for PRGs against ROBPs. More precisely, our procedure transforms any PRG against length n width w ROBP with error 1/poly(n) having seed length s to a WPRG with seed length s + O(logw/(ε) ⋅ log log1/(ε)). By instantiating our procedure with Nisan’s PRG [Noam Nisan, 1992] we obtain a WPRG with seed length O(log{n} ⋅ log(nw) + logw/(ε) ⋅ log log 1/(ε)). This improves upon [Braverman et al., 2020] and is incomparable with [Eshan Chattopadhyay and Jyun-Jie Liao, 2020]. Our construction is significantly simpler on the technical side and is conceptually cleaner. Another advantage of our construction is its low space complexity O(log{nw})+poly(log log1/(ε)) which is logarithmic in n for interesting values of the error parameter ε. Previous constructions (like [Braverman et al., 2020; Eshan Chattopadhyay and Jyun-Jie Liao, 2020]) specify the seed length but not the space complexity, though it is plausible they can also achieve such (or close) space complexity.
Cite as

Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, and Amnon Ta-Shma. Error Reduction for Weighted PRGs Against Read Once Branching Programs. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cohen_et_al:LIPIcs.CCC.2021.22,
  author =	{Cohen, Gil and Doron, Dean and Renard, Oren and Sberlo, Ori and Ta-Shma, Amnon},
  title =	{{Error Reduction for Weighted PRGs Against Read Once Branching Programs}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.22},
  URN =		{urn:nbn:de:0030-drops-142963},
  doi =		{10.4230/LIPIcs.CCC.2021.22},
  annote =	{Keywords: Pseudorandom generators, Read once branching programs, Space-bounded computation}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.111
Fourier Conjectures, Correlation Bounds, and Majority

Authors: Emanuele Viola

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Recently several conjectures were made regarding the Fourier spectrum of low-degree polynomials. We show that these conjectures imply new correlation bounds for functions related to Majority. Then we prove several new results on correlation bounds which aim to, but don't, resolve the conjectures. In particular, we prove several new results on Majority which are of independent interest and complement Smolensky’s classic result.
Cite as

Emanuele Viola. Fourier Conjectures, Correlation Bounds, and Majority. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 111:1-111:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{viola:LIPIcs.ICALP.2021.111,
  author =	{Viola, Emanuele},
  title =	{{Fourier Conjectures, Correlation Bounds, and Majority}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{111:1--111:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.111},
  URN =		{urn:nbn:de:0030-drops-141806},
  doi =		{10.4230/LIPIcs.ICALP.2021.111},
  annote =	{Keywords: Fourier analysis, polynomials, Majority, correlation, lower bound, conjectures}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.25
Optimal Error Pseudodistributions for Read-Once Branching Programs

Authors: Eshan Chattopadhyay and Jyun-Jie Liao

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length n and width w read-once branching programs with seed length O(log n⋅ log(nw)+log n⋅log(1/ε)) and error ε. It remains a central question to reduce the seed length to O(log (nw/ε)), which would prove that 𝐁𝐏𝐋 = 𝐋. However, there has been no improvement on Nisan’s construction for the case n = w, which is most relevant to space-bounded derandomization. Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit construction of a PRPD with seed length Õ(log n⋅ log(nw)+log(1/ε)). A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing 𝐁𝐏𝐋 and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length O(log n⋅ log(nw)+log(1/ε)), but their techniques are restricted to hitting sets. In this work, we construct a PRPD with seed length O(log n⋅ log (nw)⋅ log log(nw)+log(1/ε)). This improves upon the construction by Braverman, Cogen and Garg by a O(log log(1/ε)) factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.
Cite as

Eshan Chattopadhyay and Jyun-Jie Liao. Optimal Error Pseudodistributions for Read-Once Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 25:1-25:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2020.25,
  author =	{Chattopadhyay, Eshan and Liao, Jyun-Jie},
  title =	{{Optimal Error Pseudodistributions for Read-Once Branching Programs}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{25:1--25:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.25},
  URN =		{urn:nbn:de:0030-drops-125779},
  doi =		{10.4230/LIPIcs.CCC.2020.25},
  annote =	{Keywords: Derandomization, explicit constructions, space-bounded computation}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.4
Simple and Efficient Pseudorandom Generators from Gaussian Processes

Authors: Eshan Chattopadhyay, Anindya De, and Rocco A. Servedio

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
We show that a very simple pseudorandom generator fools intersections of k linear threshold functions (LTFs) and arbitrary functions of k LTFs over n-dimensional Gaussian space. The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of PRGs for functions of halfspaces. Our analysis for arbitrary functions of LTFs establishes bounds on the Wasserstein distance between Gaussian random vectors with similar covariance matrices, and combines these bounds with a conversion from Wasserstein distance to "union-of-orthants" distance from [Xi Chen et al., 2014]. Our analysis for intersections of LTFs uses extensions of the classical Sudakov-Fernique type inequalities, which give bounds on the difference between the expectations of the maxima of two Gaussian random vectors with similar covariance matrices. For all values of k, our generator has seed length O(log n) + poly(k) for arbitrary functions of k LTFs and O(log n) + poly(log k) for intersections of k LTFs. The best previous result, due to [Gopalan et al., 2010], only gave such PRGs for arbitrary functions of k LTFs when k=O(log log n) and for intersections of k LTFs when k=O((log n)/(log log n)). Thus our PRG achieves an O(log n) seed length for values of k that are exponentially larger than previous work could achieve. By combining our PRG over Gaussian space with an invariance principle for arbitrary functions of LTFs and with a regularity lemma, we obtain a deterministic algorithm that approximately counts satisfying assignments of arbitrary functions of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(k,1/epsilon)} for all values of k. This algorithm has a poly(n) runtime for k =(log n)^c for some absolute constant c>0, while the previous best poly(n)-time algorithms could only handle k = O(log log n). For intersections of LTFs, by combining these tools with a recent PRG due to [R. O'Donnell et al., 2018], we obtain a deterministic algorithm that can approximately count satisfying assignments of intersections of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(log k, 1/epsilon)}. This algorithm has a poly(n) runtime for k =2^{(log n)^c} for some absolute constant c>0, while the previous best poly(n)-time algorithms for intersections of k LTFs, due to [Gopalan et al., 2010], could only handle k=O((log n)/(log log n)).
Cite as

Eshan Chattopadhyay, Anindya De, and Rocco A. Servedio. Simple and Efficient Pseudorandom Generators from Gaussian Processes. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 4:1-4:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2019.4,
  author =	{Chattopadhyay, Eshan and De, Anindya and Servedio, Rocco A.},
  title =	{{Simple and Efficient Pseudorandom Generators from Gaussian Processes}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{4:1--4:33},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.4},
  URN =		{urn:nbn:de:0030-drops-108262},
  doi =		{10.4230/LIPIcs.CCC.2019.4},
  annote =	{Keywords: Polynomial threshold functions, Gaussian processes, Johnson-Lindenstrauss, pseudorandom generators}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.22
Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates

Authors: Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH. As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.
Cite as

Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2019.22,
  author =	{Chattopadhyay, Eshan and Hatami, Pooya and Lovett, Shachar and Tal, Avishay},
  title =	{{Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{22:1--22:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.22},
  URN =		{urn:nbn:de:0030-drops-101150},
  doi =		{10.4230/LIPIcs.ITCS.2019.22},
  annote =	{Keywords: Derandomization, Pseudorandom generator, Explicit construction, Random walk, Small-depth circuits with parity gates}
}
Document
DOI: 10.4230/LIPIcs.CCC.2018.1
Pseudorandom Generators from Polarizing Random Walks

Authors: Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett

Published in: LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

Abstract
We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [-1,1]^n. Next, we use a fractional pseudorandom generator as steps of a random walk in [-1,1]^n that converges to {-1,1}^n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.
Cite as

Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom Generators from Polarizing Random Walks. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2018.1,
  author =	{Chattopadhyay, Eshan and Hatami, Pooya and Hosseini, Kaave and Lovett, Shachar},
  title =	{{Pseudorandom Generators from Polarizing Random Walks}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{1:1--1:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Servedio, Rocco A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.1},
  URN =		{urn:nbn:de:0030-drops-88880},
  doi =		{10.4230/LIPIcs.CCC.2018.1},
  annote =	{Keywords: AC0, branching program, polarization, pseudorandom generators, random walks, Sensitivity}
}
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