Reconstruction of Real Depth-3 Circuits with Top Fan-In 2

Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In this paper we present a polynomial time randomized algorithm for reconstructing SigmaPiSigma(2) circuits over F (char(F)=0), i.e. depth-3 circuits with fan-in 2 at the top addition gate and having coefficients from a field of characteristic 0. The algorithm needs only a blackbox query access to the polynomial f in F[x_1,..., x_n] of degree d, computable by a SigmaPiSigma(2) circuit C. In addition, we assume that the "simple rank" of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time poly(n,d) and returns an equivalent SigmaPiSigma(2) circuit (with high probability). The problem of reconstructing SigmaPiSigma(2) circuits over finite fields was first proposed by Shpilka [Shpilka, STOC 2007]. The generalization to SigmaPiSigma(k) circuits, k = O(1) (over finite fields) was addressed by Karnin and Shpilka in [Karnin/Shpilka, CCC 2015]. The techniques in these previous involve iterating over all objects of certain kinds over the ambient field and thus the running time depends on the size of the field F. Their reconstruction algorithm uses lower bounds on the lengths of Linear Locally Decodable Codes with 2 queries. In our settings, such ideas immediately pose a problem and we need new ideas to handle the case of the characteristic 0 field F. Our main techniques are based on the use of Quantitative Sylvester Gallai Theorems from the work of Barak et al. [Barak/Dvir/Wigderson/Yehudayoff, STOC 2011] to find a small collection of "nice" subspaces to project onto. The heart of our paper lies in subtle applications of the Quantitative Sylvester Gallai theorems to prove why projections w.r.t. the "nice" subspaces can be "glued". We also use Brill\u27s Equations from [Gelfand/Kapranov/Zelevinsky, 1994] to construct a small set of candidate linear forms (containing linear forms from both gates). Another important technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [Kaltofen/Trager, J. Symb. Comp. 1990]

Parallel dynamics and computational complexity of the Bak-Sneppen model

The parallel computational complexity of the Bak-Sneppen evolution model is studied. It is shown that Bak-Sneppen histories can be generated by a massively parallel computer in a time that is polylogarithmic in the length of the history. In this parallel dynamics, histories are built up via a nested hierarchy of avalanches. Stated in another way, the main result is that the logical depth of producing a Bak-Sneppen history is exponentially less than the length of the history. This finding is surprising because the self-organized critical state of the Bak-Sneppen model has long range correlations in time and space that appear to imply that the dynamics is sequential and history dependent. The parallel dynamics for generating Bak-Sneppen histories is contrasted to standard Bak-Sneppen dynamics. Standard dynamics and an alternate method for generating histories, conditional dynamics, are both shown to be related to P-complete natural decision problems implying that they cannot be efficiently implemented in parallel.Comment: 37 pages, 12 figure

Micro Fourier Transform Profilometry (μ\muFTP): 3D shape measurement at 10,000 frames per second

Recent advances in imaging sensors and digital light projection technology have facilitated a rapid progress in 3D optical sensing, enabling 3D surfaces of complex-shaped objects to be captured with improved resolution and accuracy. However, due to the large number of projection patterns required for phase recovery and disambiguation, the maximum fame rates of current 3D shape measurement techniques are still limited to the range of hundreds of frames per second (fps). Here, we demonstrate a new 3D dynamic imaging technique, Micro Fourier Transform Profilometry ($\mu$FTP), which can capture 3D surfaces of transient events at up to 10,000 fps based on our newly developed high-speed fringe projection system. Compared with existing techniques, $\mu$FTP has the prominent advantage of recovering an accurate, unambiguous, and dense 3D point cloud with only two projected patterns. Furthermore, the phase information is encoded within a single high-frequency fringe image, thereby allowing motion-artifact-free reconstruction of transient events with temporal resolution of 50 microseconds. To show $\mu$FTP's broad utility, we use it to reconstruct 3D videos of 4 transient scenes: vibrating cantilevers, rotating fan blades, bullet fired from a toy gun, and balloon's explosion triggered by a flying dart, which were previously difficult or even unable to be captured with conventional approaches.Comment: This manuscript was originally submitted on 30th January 1

Discovering the roots: Uniform closure results for algebraic classes under factoring

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{\Omega(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$'s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{\Omega(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $\tau$, $f(\tau\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

Practical recommendations for gradient-based training of deep architectures

Learning algorithms related to artificial neural networks and in particular for Deep Learning may seem to involve many bells and whistles, called hyper-parameters. This chapter is meant as a practical guide with recommendations for some of the most commonly used hyper-parameters, in particular in the context of learning algorithms based on back-propagated gradient and gradient-based optimization. It also discusses how to deal with the fact that more interesting results can be obtained when allowing one to adjust many hyper-parameters. Overall, it describes elements of the practice used to successfully and efficiently train and debug large-scale and often deep multi-layer neural networks. It closes with open questions about the training difficulties observed with deeper architectures