FOCS 2011
TechTalks from event: FOCS 2011
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11A

Information Equals Amortized CommunicationWe show how to efficiently simulate the sending of a message $M$ to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver. This is a generalization and strengthening of the SlepianWolf theorem, which shows how to carry out such a simulation with low \emph{amortized} communication in the case that $M$ is a deterministic function of $X$. A caveat is that our simulation is interactive. As a consequence, we obtain new relationships between the randomized amortized communication complexity of a function, and its information complexity. We prove that for any given distribution on inputs, the internal information cost (namely the information revealed to the parties) involved in computing any relation or function using a two party interactive protocol is {\em exactly} equal to the amortized communication complexity of computing independent copies of the same relation or function. Here by amortized communication complexity we mean the average per copy communication in the best protocol for computing multiple copies, with a bound on the error in each copy (i.e.\ we require only that the output in each coordinate is correct with good probability, and do not require that all outputs are simultaneously correct). This significantly simplifies the relationships between the various measures of complexity for average case communication protocols, and proves that if a function's information cost is smaller than its communication complexity, then multiple copies of the function can be computed more efficiently in parallel than sequentially. Finally, we show that the only way to prove a strong direct sum theorem for randomized communication complexity is by solving a particular variant of the pointer jumping problem that we define. If this problem has a cheap communication protocol, then a strong direct sum theorem must hold. On the other hand, if it does not, then the problem itself gives a counterexample for the direct sum question. In the process we show that a strong direct sum theorem for communication complexity holds if and only if efficient compression of communication protocols is possible.

Delays and the Capacity of Continuoustime ChannelsAny physical channel of communication offers two potential reasons why its capacity (the number of bits it can transmit in a unit of time) might be unbounded: (1) (Uncountably) infinitely many choices of signal strength at any given time, and (2) (Uncountably) infinitely many instances of time at which signals may be sent. However channel noise cancels out the potential unboundedness of the first aspect, leaving typical channels with only a finite capacity per instant of time. The latter source of infinity seems less extensively studied. A potential source of unreliability that might restrict the capacity also from the second aspect is ``delay'': Signals transmitted by the sender at a given point of time may not be received with a predictable delay at the receiving end. In this work we examine this source of uncertainty by considering a simple discrete model of delay errors. In our model the communicating parties get to subdivide time as finely as they wish, but still have to cope with communication delays that are variable. The continuous process becomes the limit of our process as the time subdivision becomes infinitesimal. We analyze the limits of such channels and reach somewhat surprising conclusions: The capacity of a physical channel is finitely bounded only if at least one of the two sources of error (signal noise or delay noise) is adversarial. If both error sources are stochastic, or the adversarial source is noise that is independent of the stochastic delay, the capacity of the associated physical channel is infinite!

Efficient and Explicit Coding for Interactive CommunicationIn this work, we study the fundamental problem of reliable interactive communication over a noisy channel. In a breakthrough sequence of papers published in 1992 and 1993, Schulman gave nonconstructive proofs of the existence of general methods to emulate any twoparty interactive protocol such that: (1) the emulation protocol only takes a constantfactor longer than the original protocol, and (2) if the emulation protocol is executed over any discrete memoryless noisy channel with constant capacity, then the probability that the emulation protocol fails to perfectly emulate the original protocol is exponentially small in the total length of the protocol. Unfortunately, Schulman's emulation procedures either only work in a nonstandard model with a large amount of shared randomness, or are nonconstructive in that they rely on the existence of "absolute" tree codes. The only known proofs of the existence of absolute tree codes are nonconstructive, and finding an explicit construction remains an important open problem. Indeed, randomly generated tree codes are not absolute tree codes with overwhelming probability. In this work, we revisit the problem of reliable interactive communication, and obtain the first fully explicit (randomized) efficient constantrate emulation procedure for reliable interactive communication. Our protocol works for any discrete memoryless noisy channel with constant capacity, and our protocol's probability of failure is exponentially small in the total length of the protocol. We accomplish this goal by obtaining the following results: We introduce a new notion of goodness for a tree code, and define the notion of a potent tree code. We believe that this notion is of independent interest. We prove the correctness of an explicit emulation procedure based on any potent tree code. (This replaces the need for absolute tree codes in the work of Schulman.) We show that a randomly generated tree code (with suitable constant alphabet size) is an efficiently decodable potent tree code with overwhelming probability. Furthermore we are able to partially derandomize this result by means of epsilonbiased distributions using only $O(n)$ random bits, where $n$ is the depth of the tree.These (derandomized) results allow us to obtain our main result. Our results also extend to the case of interactive multiparty communication among a constant number of parties.

Efficient Reconstruction of Random Multilinear FormulasIn the reconstruction problem for a multivariate polynomial $f$, we have blackbox access to $f$ and the goal is to efficiently reconstruct a representation of $f$ in a suitable model of computation. We give a polynomial time randomized algorithm for reconstructing random multilinear formulas. Our algorithm succeeds with high probability when given blackbox access to the polynomial computed by a random multilinear formula according to a natural distribution. This is the strongest model of computation for which a reconstruction algorithm is presently known, albeit efficient in a distributional sense rather than in the worstcase. Previous results on this problem considered much weaker models such as depth3 circuits with various restrictions or readonce formulas. Our proof uses ranks of partial derivative matrices as a key ingredient and combines it with analysis of the algebraic structure of random multilinear formulas. Partial derivative matrices have earlier been used to prove lower bounds in a number of models of arithmetic complexity, including multilinear formulas and constant depth circuits. As such, our results give supporting evidence to the general thesis that mathematical properties that capture efficient computation in a model should also enable learning algorithms for functions efficiently computable in that model.

New extension of the Weil bound for character sums with applications to codingThe Weil bound for character sums is a deep result in Algebraic Geometry with many applications both in mathematics and in the theoretical computer science. The Weil bound states that for any polynomial $f(x)$ over a finite field $\mathbb{F}$ and any additive character $\chi:\mathbb{F} \to \mathbb{C}$, either $\chi(f(x))$ is a constant function or it is distributed close to uniform. The Weil bound is quite effective as long as $\deg(f) \ll \sqrt{\mathbb{F}}$, but it breaks down when the degree of $f$ exceeds $\sqrt{\mathbb{F}}$. As the Weil bound plays a central role in many areas, finding extensions for polynomials of larger degree is an important problem with many possible applications. In this work we develop such an extension over finite fields $\mathbb{F}_{p^n}$ of small characteristic: we prove that if $f(x)=g(x)+h(x)$ where $\deg(g) \ll \sqrt{\mathbb{F}}$ and $h(x)$ is a sparse polynomial of arbitrary degree but bounded weight degree, then the same conclusion of the classical Weil bound still holds: either $\chi(f(x))$ is constant or its distribution is close to uniform. In particular, this shows that the subcode of ReedMuller codes of degree $\omega(1)$ generated by traces of sparse polynomials is a code with near optimal distance, while ReedMuller of such a degree has no distance (i.e. $o(1)$ distance) ; this is one of the few examples where one can prove that sparse polynomials behave differently from nonsparse polynomials of the same degree. As an application we prove new general results for affine invariant codes. We prove that any affineinvariant subspace of quasipolynomial size is (1) indeed a code (i.e. has good distance) and (2) is locally testable. Previous results for general affine invariant codes were known only for codes of polynomial size, and of length $2^n$ where $n$ needed to be a prime. Thus, our techniques are the first to extend to general families of such codes of superpolynomial size, where we also remove the requirement from $n$ to be a prime. The proof is based on two main ingredients: the extension of the Weil bound for character sums, and a new Fourieranalytic approach for estimating the weight distribution of general codes with large dual distance, which may be of independent interest.