[Part 1] [Part 2]

Abstract: 

This will be a tutorial about applications of polynomials and multiplicities to codes and extractors. I will talk about the list-decoding of Reed-Solomon codes, analyzing Kakeya sets, and about multiplicity codes.

Title: Polynomials and Multiplicities II
Speaker: Swastik Kopparty, Rutgers University

[Complete Video]

Abstract: 

This will be a continuation of the tutorial. I will give the Stepanov-Schmidt elementary proof of the Weil bound and a recent explicit construction (joint with Venkatesan Guruswami) of subspace designs.

[Complete Video]

Abstract: 

Many off-the-shelf algorithms for compressed sensing, like linear programming, rely on the compressed sensing matrix satisfying a sufficient condition called the “Restricted Isometry Property” (RIP). These algorithms also run faster if the compressed sensing matrix supports fast matrix-vector multiplication—for example, if it consists of random rows of a DFT. However, showing that the Fourier ensemble has the RIP with an optimal number of measurements appears to be a very difficult problem. In this talk, I will present some progress: we show that a matrix whose rows are sparse linear combinations of Fourier measurements has the RIP with fewer rows than previous constructions. Our results give RIP matrices supporting fast multiplication with fewer rows than previously known, and also imply improved fast Johnson-Lindenstrauss transforms.

This is joint work with Eric Price (MIT) and Mary Wootters (UMich).

[Complete Video]

Abstract: 

The study of combinatorial problems with a submodular objective has attracted much attention in recent years, and is partly motivated by the fact that such problems arise in diverse settings such as combinatorial optimization, algorithmic game theory, economics, machine learning and social networks.

We present improved approximation guarantees for submodular maximization problems in the constrained and the unconstrained cases. In the constrained case we present a single unified algorithm that computes fractional solutions for the multilinear relaxation for both monotone and non-monotone objectives. This algorithm yields information-theoretic tight approximations for the Submodular-Welfare and Submodular-Max-SAT problems. Additionally, it also provides an improved (1/e)-approximation for maximizing a non-monotone submodular function over a matroid. For the unconstrained case we present a simple linear-time algorithm that achieves an information-theoretic tight approximation of 1/2.

Based on joint works with Niv Buchbinder, Moran Feldman and Joseph (Seffi) Naor.

Title: Analysis and Synthesis Methods in Compressed Sensing
Speaker: Deanna Needell, Claremont McKenna College
[Complete Video]

Abstract: In this talk we will discuss results for robust signal reconstruction from random observations via synthesis and analysis methods.  Synthesis methods attempt to identify the low-dimensional representation of the signal directly, whereas analysis type methods reconstruct in signal space. We also include provable near-optimal reconstruction guarantees for total-variation minimization using properties of the bivariate Haar transform.

Title: Finding Cliques in Large Random Graphs, Sparse PCA and Iterative Thresholding
Speaker: Andrea Montanari, Stanford
[Complete Video]

Abstract: Consider a n times n symmetric matrix A, whose entries are i.i.d. with a common law P_0, except for a small k times k submatrix whose entries are also i.i.d. but with a different law P_1. We consider the case in which P_1 and P_0 have different mean, and investigate the problem of identifying the 'anomalous' k by k submatrix. A special case of this question is the well studied problem of identifying a clique in an Erdos-Renyi random graph. A simple counting argument shows that such a submatrix can be uniquely identified via exhaustive enumeration provided k is larger than C*log(n), for a suitable constant C. Unfortunately, no practical algorithm is successful for k smaller than square root of n. Worse than this, the best available guarantees are for vanilla PCA which does not make use of the sparsity of the submatrix. I will describe an iterative thresholding algorithm that provably improves over simple PCA, and in particular finds cliques of size square root of n/e.

Title: Faster Algorithms for the Sparse Fourier Transform
Speaker: Piotr Indyk, MIT
[Slides (PDF)]

Abstract: The Fast Fourier Transform (FFT) is one of the most fundamental numerical algorithms. It computes the Discrete Fourier Transform (DFT) of an n-dimensional signal in O(n log n) time. The algorithm plays an important role in many areas. It is not known whether its running time can be improved. However, in many applications, most of the Fourier coefficients of a signal are "small" or equal to zero, i.e., the output of the transform is (approximately) sparse. In this case, it is known that one can compute the set of non-zero coefficients faster than in O(n log n) time.

[Complete Video]

Abstract: 

We give a polynomial time algorithm for the lossy population recovery problem. In this problem, the goal is to approximately learn an unknown distribution on binary strings of length n from lossy samples: for some parameter μ each coordinate of the sample is preserved with probability μ and otherwise is replaced by a ‘?’.

The running time and number of samples needed for our algorithm is polynomial in n and 1/ε for each fixed μ > 0. This improves on algorithm of Wigderson and Yehudayoff that runs in quasi-polynomial time for any μ > 0 and the polynomial time algorithm of Dvir et al which was shown to work for μ > 0.30 by Batman et al. In fact, our algorithm also works in the more general framework of Batman et al. in which there is no a priori bound on the size of the support of the distribution.

The algorithm we analyze is implicit in previous work; our main contribution is to analyze the algorithm by showing (via linear programming duality and connections to complex analysis) that a certain matrix associated with the problem has a robust local inverse even though its condition number is exponentially small. A corollary of our result is the first polynomial time algorithm for learning DNFs in the restriction access model of Dvir et al and hence this model joins the random walk model of Bshouty et al as the only examples of passive learning in which DNFs can be learned in polynomial time.

Joint work with Mike Saks

Description: The workshop will bring together researchers from computer science, mathematics, physics, biology, and engineering to explore interactions among algorithms, dynamical systems, statistical physics, and complexity theory (in all senses of the term).

Dates: The workshop will go from May 20 to May 21 and will be held at Nassau Inn in Princeton, NJ.

Tentative Program/Schedule:

Monday May 20 

9:15-10:00 – Breakfast and registration

10:00-10:30 – Opening remarks 
10:30-11:00 – Leslie Valiant  "Algorithms for adaptive phenomena"

11:00-11:30 – Break 
11:30-12:00 – Simon Levin  "Bounded rationality and decision-making"
12:00-12:30 – Cris Moore "Universality, hardness, engineering, and messiness"

12:30-2:00   Lunch 

2:00-2:30 – Albert Libchaber  "Experiments on artificial cells"
2:30-3:00 – Jack Lutz  "Parametrizing self-assembly"
3:00-3:30 – Break 
3:30-4:00 – Naomi Leonard "Network topology and the evolution of leadership in collective migration"
4:00-4:30 – Mark Braverman  "Noise versus computational intractability in dynamics"

Evening Banquet 

Tuesday May 21 

9:15-10:00 – Breakfast

10:00-10:30 – Luca Cardelli  "The cell cycle switch computes approximate majority"
10:30-11:00 – Phil Holmes  "The neuro-mechanics of running cockroaches: How much natural detail do we need?"
11:00-11:30 – Break 
11:30-12:00 – Ishanu Chattopadhyay "Data smashing: Finding causal similarity in natural data sets" 

12:00-12:30 – Joel Lebowitz  "Microscopic models of macroscopic behavior"

12:30-2:00   Lunch 

2:00-2:30 – Ofer Feinerman "Fighting noise with limited resources: an ant colony perspective"
2:30-3:00 - Rebecca Schulman  "Universal molecular algorithms for learning and pattern formation"

3:00-3:30 – Break 

3:30-4:00 – Eduardo Sontag  "Some invariant aspects of dynamical behavior of cell signaling pathways"
4:00-4:30 – Amos Korman "Integrating theoretical algorithmic ideas in empirical biological study"

Hotel: If you plan to come to the workshop and need a hotel room, we suggest that you reserve a room at the Nassau Inn as soon as possible. Their phone number is 1-800-862-7728. 

cforms contact form by delicious:days

[Complete Video] (note: audio cuts out for roughly 7 minutes at around 00:39:30)

Abstract: 

Much as probabilistic thinking did starting in the early 80s, quantum computing is expanding the core questions of complexity theory in fundamental new directions. For example, here is a list of three basic questions about quantum mechanics that are at their heart questions about computational complexity:

1. Do `typical’ quantum states that occur in Nature have succinct (polynomial) description?
2. Can quantum systems at room temperature exhibit exponential complexity?
3. Is the scientific method sufficiently powerful to comprehend general quantum systems?

Each of these issues is best studied through the computational lens as a question about computation. The resulting questions lie at the core of computational complexity theory. The first asks about the structure of solutions to the quantum analog of SAT. The second asks whether there is a quantum analog of the PCP theorem. And the third can be formulated as a question about interactive proof systems with quantum polynomial time provers. I will briefly outline these connections and the state of the art on these questions.

[Complete Video]

Abstract: 

The EQUALITY problem — where Alice and Bob must decide if their respective inputs are equal — is usually one’s first encounter with communication complexity. Its deterministic and randomized complexity were settled decades ago, but we find several new things to say by considering two subtle aspects. First, we study the expected communication cost (at a worst-case input) for a protocol that uses limited interaction. Second, we study the information cost of such protocols. We obtain asymptotically optimal rounds-versus-communication and rounds-versus-information tradeoffs for EQUALITY.

As an application of our information cost bounds, we obtain new, and asymptotically optimal, bounded-round randomized lower bounds for OR-EQUALITY and k-DISJOINTNESS (where input sets have size ≤ k).

Joint work with Joshua Brody (Aarhus) and Ranganath Kondapally (Dartmouth).

[Part 1] [Part 2]

Abstract: 

In 1953 Grothendieck proved an inequality of immense importance to several areas of mathematics, as well as having interesting applications to combinatorial optimization. In the same paper Grothendieck also conjectured the validity of a “noncommutative” version of his inequality, a conjecture that was solved affirmatively by Pisier in 1978, yielding an important tool for the study of C* algebras. This talk is devoted to a description of algorithmic implications of the noncommutative Grothendieck inequality. The talk will be self-contained and does not assume any background other than basic linear algebra and probability. We will start with a quick description of the classical Grothendieck inequality, explain how it leads to its noncommutative counterpart, and present a proof of the inequality that is an interesting rounding algorithm for a certain semidefinite program. The efficiency of this algorithm yields several algorithmic applications that we will describe, including the first constant factor approximation algorithms for the orthogonal Procrustes problem and robust versions of principle component analysis (PCA).

Joint work with Oded Regev and Thomas Vidick.

Title: Grothendieck Inequalities in Quantum Games and a Proof of the Operator Space Grothendieck Inequality
Speaker: Oded Regev, Courant Institute, New York University

[Complete Video]

Abstract: 

One of the most striking consequences of quantum mechanics (dating back to Einstein et al.) is that two remote parties who share entanglement can exhibit correlations that are impossible without entanglement. I will describe the history of this problem and its connection (mainly due to Tsirelson) to Grothendieck’s inequality. I will then describe our recent work on quantum XOR games and discuss their connection to the non-commutative Grothendieck inequality and to the very recent operator-space Grothendieck inequality.

We will then discuss our recent new proof of the operator-space Grothendieck inequality, which is based on intuitions coming from quantum information and computer science.

Based on two papers with Thomas Vidick.

Video: [Part 1] [Part 2]

Abstract: 

In this talk, I will discuss trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work. For a set of d linear queries over a database x ∈ ℝ^N, we seek to find the differentially private mechanism that has the minimum mean squared error. I will first describe a O(log² d) approximation algorithm for this problem, for the case of (ε,δ)-differential privacy. The mechanism is simple, efficient and adds correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex geometry.

We will next consider this question in the case when the number of queries exceeds the number of individuals in the database, i.e. when d > n = |x|₁. It is known that better mechanisms exist in this setting. I will then describe an (ε,δ)-differentially private mechanism which is optimal up to a polylog(d,N) factor for any given query set A and any given upper bound n on |x|₁. This approximation is achieved by coupling the Gaussian noise addition approach with a linear regression step. We give an analogous result for the ε-differential privacy setting. We also improve on the mean squared error upper bound for answering counting queries on a database of size n by Blum, Ligett, and Roth, and match the lower bound implied by the work of Dinur and Nissim up to logarithmic factors.

The connection between hereditary discrepancy and the privacy mechanism also enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix A.

This talk is based on joint work with Alex Nikolov and Li Zhang.