Negation-Limited Formulas

We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas. We give an efficient transformation of formulas with t negation gates to circuits with log(t) negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC\u2715), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations. In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas

Which groups are amenable to proving exponent two for matrix multiplication?

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $\omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $\omega$ and methods for ruling them out.Comment: 23 pages, 1 figur

Origin of conductivity cross over in entangled multi-walled carbon nanotube network filled by iron

A realistic transport model showing the interplay of the hopping transport between the outer shells of iron filled entangled multi-walled carbon nanotubes (MWNT) and the diffusive transport through the inner part of the tubes, as a function of the filling percentage, is developed. This model is based on low-temperature electrical resistivity and magneto-resistance (MR) measurements. The conductivity at low temperatures showed a crossover from Efros-Shklovski (E-S) variable range hopping (VRH) to Mott VRH in 3 dimensions (3D) between the neighboring tubes as the iron weight percentage is increased from 11% to 19% in the MWNTs. The MR in the hopping regime is strongly dependent on temperature as well as magnetic field and shows both positive and negative signs, which are discussed in terms of wave function shrinkage and quantum interference effects, respectively. A further increase of the iron percentage from 19% to 31% gives a conductivity crossover from Mott VRH to 3D weak localization (WL). This change is ascribed to the formation of long iron nanowires at the core of the nanotubes, which yields a long dephasing length (e.g. 30 nm) at the lowest measured temperature. Although the overall transport in this network is described by a 3D WL model, the weak temperature dependence of inelastic scattering length expressed as L_phi ~T^-0.3 suggests the possibility for the presence of one-dimensional channels in the network due to the formation of long Fe nanowires inside the tubes, which might introduce an alignment in the random structure.Comment: 29 pages,10 figures, 2 tables, submitted to Phys. Rev.

Strain Analysis of Initial Stage Sintering of 316L SS Three Dimensionally Printed (3DP TM) Components

Mechanical Engineerin

Bayesian Compression for Deep Learning

Compression and computational efficiency in deep learning have become a problem of great significance. In this work, we argue that the most principled and effective way to attack this problem is by adopting a Bayesian point of view, where through sparsity inducing priors we prune large parts of the network. We introduce two novelties in this paper: 1) we use hierarchical priors to prune nodes instead of individual weights, and 2) we use the posterior uncertainties to determine the optimal fixed point precision to encode the weights. Both factors significantly contribute to achieving the state of the art in terms of compression rates, while still staying competitive with methods designed to optimize for speed or energy efficiency.Comment: Published as a conference paper at NIPS 201