On the power of homogeneous depth 4 arithmetic circuits

We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$. Our results hold for the {\it Iterated Matrix Multiplication} polynomial - in particular we show that any homogeneous depth 4 circuit computing the $(1,1)$ entry in the product of $n$ generic matrices of dimension $n^{O(1)}$ must have size $n^{\Omega(\sqrt{n})}$. Our results strengthen previous works in two significant ways. Our lower bounds hold for a polynomial in $VP$. Prior to our work, Kayal et al [KLSS14] proved an exponential lower bound for homogeneous depth 4 circuits (over fields of characteristic zero) computing a poly in $VNP$. The best known lower bounds for a depth 4 homogeneous circuit computing a poly in $VP$ was the bound of $n^{\Omega(\log n)}$ by [LSS, KLSS14].Our exponential lower bounds also give the first exponential separation between general arithmetic circuits and homogeneous depth 4 arithmetic circuits. In particular they imply that the depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even for reductions to general homogeneous depth 4 circuits (without the restriction of bounded bottom fanin). Our lower bound holds over all fields. The lower bound of [KLSS14] worked only over fields of characteristic zero. Prior to our work, the best lower bound for homogeneous depth 4 circuits over fields of positive characteristic was $n^{\Omega(\log n)}$ [LSS, KLSS14]

Sums of products of polynomials in few variables : lower bounds and polynomial identity testing

We study the complexity of representing polynomials as a sum of products of polynomials in few variables. More precisely, we study representations of the form $$P = \sum_{i = 1}^T \prod_{j = 1}^d Q_{ij}$$ such that each $Q_{ij}$ is an arbitrary polynomial that depends on at most $s$ variables. We prove the following results. 1. Over fields of characteristic zero, for every constant $\mu$ such that $0 \leq \mu < 1$, we give an explicit family of polynomials $\{P_{N}\}$, where $P_{N}$ is of degree $n$ in $N = n^{O(1)}$ variables, such that any representation of the above type for $P_{N}$ with $s = N^{\mu}$ requires $Td \geq n^{\Omega(\sqrt{n})}$. This strengthens a recent result of Kayal and Saha [KS14a] which showed similar lower bounds for the model of sums of products of linear forms in few variables. It is known that any asymptotic improvement in the exponent of the lower bounds (even for $s = \sqrt{n}$) would separate VP and VNP[KS14a]. 2. We obtain a deterministic subexponential time blackbox polynomial identity testing (PIT) algorithm for circuits computed by the above model when $T$ and the individual degree of each variable in $P$ are at most $\log^{O(1)} N$ and $s \leq N^{\mu}$ for any constant $\mu < 1/2$. We get quasipolynomial running time when $s < \log^{O(1)} N$. The PIT algorithm is obtained by combining our lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04]. To the best of our knowledge, this is the first nontrivial PIT algorithm for this model (even for the case $s=2$), and the first nontrivial PIT algorithm obtained from lower bounds for small depth circuits

Determining General and Specific Purpose Transfers : An Integrated Approach

This study attempts to provide an alternative methodology to calculate the horizontal equalization transfers. This methodology follows the Australian horizontal equalization principle using a panel model methodology where both revenue and expenditure side considerations are involved. First, it applies the Canadian model in calculating the fiscal capacity equalization. Then the expenditure side equalization has been carried out for two services - education and health. Results of the exercise indicate that the transfers suggested by the panel model are more progressive than the TFC recommended transfers.horizontal equalization transfers, education, health, fiscal capacity equalization

Improved rank bounds for design matrices and a new proof of Kelly's theorem

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in which they were used to answer questions regarding point configurations. In this work we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai theorem