A probabilistic remark on algebraic program testing

Counting Real Roots in Polynomial-Time via Diophantine Approximation

Suppose A = { a 1 , … , a n + 2 } ⊂ Z n has cardinality n + 2 , with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = ( f 1 , … , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log ( d H ) . We also discuss a number-theoretic hypothesis that would imply a further speed-up to time polynomial in n as well.

Bootstrapping variables in algebraic circuits

We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One needs only to consider size-s degree-s circuits that depend on the first [Formula: see text] variables (where c is a constant and composes a logarithm with itself c times). Thus, the hitting-set generator (hsg) manifests a bootstrapping behavior-a partial hsg against very few variables can be efficiently grown to a complete hsg. A Boolean analog, or a pseudorandom generator property of this type, is unheard of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w.r.t. variables. This is repeated c times in an efficient way. Pushing the envelope further we show that (i) a quadratic-time blackbox PIT for 6,913-variate degree-s size-s polynomials will lead to a "near"-complete derandomization of PIT and (ii) a blackbox PIT for n-variate degree-s size-s circuits in [Formula: see text] time, for [Formula: see text], will lead to a near-complete derandomization of PIT (in contrast, [Formula: see text] time is trivial). Our second idea is to study depth-4 circuits that depend on constantly many variables. We show that a polynomial-time computable, [Formula: see text]-degree hsg for trivariate depth-4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of Kabanets and Impagliazzo [Kabanets V, Impagliazzo R (2003) Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing STOC '03].

On facilitating the use of HARDI in population studies by creating rotation-invariant markers

We design and evaluate a novel method to compute rotationally invariant features using High Angular Resolution Diffusion Imaging (HARDI) data. These measures quantify the complexity of the angular diffusion profile modeled using a higher order model, thereby giving more information than classical diffusion tensor-derived parameters. The method is based on the spherical harmonic (SH) representation of the angular diffusion information, and is generalizable to a range of HARDI reconstruction models. These scalars are obtained as homogeneous polynomials of the SH representation of a HARDI reconstruction model. We show that finding such polynomials is equivalent to solving a large linear system of equations, and present a numerical method based on sparse matrices to efficiently solve this system. Among the solutions, we only keep a subset of algebraically independent polynomials, using an algorithm based on a numerical implementation of the Jacobian criterion. We compute a set of 12 or 25 rotationally invariant measures representative of the underlying white matter for the rank-4 or rank-6 spherical harmonics (SHs) representation of the apparent diffusion coefficient (ADC) profile, respectively. Synthetic data was used to investigate and quantify the difference in contrast. Real data acquired with multiple repetitions showed that within subject variation in the invariants was less than the difference across subjects - facilitating their use to study population differences. These results demonstrate that our measures are able to characterize white matter, especially complex white matter found in regions of fiber crossings and hence can be used to derive new biomarkers for HARDI and can be used for HARDI-based population analysis.