Decidability of Non-interactive Simulation of Joint Distributions

We present decidability results for a sub-class of "non-interactive" simulation problems, a well-studied class of problems in information theory. A non-interactive simulation problem is specified by two distributions P(x, y) and Q(u, v): The goal is to determine if two players, Alice and Bob, that observe sequences Xn and Yn respectively where {(Xi, Yi)}ni = 1 are drawn i.i.d. from P(x, y) can generate pairs U and V respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to Q(u, v). Even when P and Q are extremely simple: e.g., P is uniform on the triples (0, 0), (0,1), (1,0) and Q is a "doubly symmetric binary source", i.e., U and V are uniform ± 1 variables with correlation say 0.49, it is open if P can simulate Q. In this work, we show that whenever P is a distribution on a finite domain and Q is a 2 × 2 distribution, then the non-interactive simulation problem is decidable: specifically, given δ > 0 the algorithm runs in time bounded by some function of P and δ and either gives a non-interactive simulation protocol that is δ-close to Q or asserts that no protocol gets O(δ)-close to Q. The main challenge to such a result is determining explicit (computable) convergence bounds on the number n of samples that need to be drawn from P(x, y) to get δ-close to Q. We invoke contemporary results from the analysis of Boolean functions such as the invariance principle and a regularity lemma to obtain such explicit bounds.

[1]  Venkatesan Guruswami,et al.  Communication With Imperfectly Shared Randomness , 2014, IEEE Transactions on Information Theory.

[2]  Renato Renner,et al.  Simple and Tight Bounds for Information Reconciliation and Privacy Amplification , 2005, ASIACRYPT.

[3]  Reza Modarres,et al.  Measures of Dependence , 2011, International Encyclopedia of Statistical Science.

[4]  Rocco A. Servedio,et al.  A Regularity Lemma, and Low-Weight Approximators, for Low-Degree Polynomial Threshold Functions , 2009, 2010 IEEE 25th Annual Conference on Computational Complexity.

[5]  Keiji Matsumoto,et al.  Entangled Games are Hard to Approximate , 2007, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[6]  Venkat Anantharam,et al.  On Non-Interactive Simulation of Joint Distributions , 2015, IEEE Transactions on Information Theory.

[7]  H. Hirschfeld A Connection between Correlation and Contingency , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  H. Gebelein Das statistische Problem der Korrelation als Variations‐ und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung , 1941 .

[9]  Salman Beigi,et al.  A New Quantum Data Processing Inequality , 2012, ArXiv.

[10]  Ryan O'Donnell,et al.  Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs? , 2007, SIAM J. Comput..

[11]  H. Witsenhausen ON SEQUENCES OF PAIRS OF DEPENDENT RANDOM VARIABLES , 1975 .

[12]  Ryan O'Donnell,et al.  Noise stability of functions with low influences: Invariance and optimality , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[13]  Salman Beigi,et al.  Impossibility of Local State Transformation via Hypercontractivity , 2013, ArXiv.

[14]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

[15]  Ryan O'Donnell,et al.  Coin flipping from a cosmic source: On error correction of truly random bits , 2004, Random Struct. Algorithms.

[16]  Aaron D. Wyner,et al.  The common information of two dependent random variables , 1975, IEEE Trans. Inf. Theory.

[17]  M. Nielsen Conditions for a Class of Entanglement Transformations , 1998, quant-ph/9811053.

[18]  Runyao Duan,et al.  Tripartite entanglement transformations and tensor rank. , 2008, Physical review letters.

[19]  A. Dembo,et al.  On the Maximum Correlation Coefficient , 2005 .

[20]  Tsuyoshi Ito,et al.  On the Role of Shared Randomness in Simultaneous Communication , 2014, ICALP.

[21]  Rocco A. Servedio,et al.  Testing Halfspaces , 2007, SIAM J. Comput..

[22]  M. Maurer,et al.  Secret Key Agreement by Public Discussion from Common Information , 2004 .

[23]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[24]  A. Peres All the Bell Inequalities , 1998, quant-ph/9807017.

[25]  Amin Gohari,et al.  On the duality of additivity and tensorization , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[26]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[27]  Elchanan Mossel,et al.  Maximally stable Gaussian partitions with discrete applications , 2009, 0903.3362.

[28]  Rudolf Ahlswede,et al.  Common Randomness in Information Theory and Cryptography - Part II: CR Capacity , 1998, IEEE Trans. Inf. Theory.

[29]  Johan Håstad,et al.  Randomly supported independence and resistance , 2009, STOC '09.

[30]  Elchanan Mossel,et al.  Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality , 2004, math/0410560.

[31]  Mark Braverman,et al.  Information Equals Amortized Communication , 2011, IEEE Transactions on Information Theory.

[32]  Rudolf Ahlswede,et al.  Common randomness in information theory and cryptography - I: Secret sharing , 1993, IEEE Trans. Inf. Theory.

[33]  Gilles Brassard,et al.  Secret-Key Reconciliation by Public Discussion , 1994, EUROCRYPT.

[34]  Prasad Raghavendra,et al.  Reductions between Expansion Problems , 2010, 2012 IEEE 27th Conference on Computational Complexity.

[35]  Imre Csiszár,et al.  Common randomness and secret key generation with a helper , 2000, IEEE Trans. Inf. Theory.

[36]  Noga Alon,et al.  The Shannon capacity of a graph and the independence numbers of its powers , 2006, IEEE Transactions on Information Theory.

[37]  Mark Braverman,et al.  Information Complexity Is Computable , 2016, ICALP.

[38]  Elchanan Mossel,et al.  Standard simplices and pluralities are not the most noise stable , 2014, ITCS.

[39]  V. Rich Personal communication , 1989, Nature.

[40]  Madhu Sudan,et al.  Communication Complexity of Permutation-Invariant Functions , 2016, SODA.

[41]  P. Wolff,et al.  Hypercontractivity of simple random variables , 2007 .

[42]  Elchanan Mossel Gaussian Bounds for Noise Correlation of Functions , 2007, FOCS 2007.

[43]  Aaron D. Wyner,et al.  The Zero Error Capacity of a Noisy Channel , 1993 .

[44]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[45]  C. Borell Geometric bounds on the Ornstein-Uhlenbeck velocity process , 1985 .

[46]  Venkat Anantharam,et al.  Non-interactive simulation of joint distributions: The Hirschfeld-Gebelein-Rényi maximal correlation and the hypercontractivity ribbon , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).