Finer Separations Between Shallow Arithmetic Circuits
نویسندگان
چکیده
In this paper, we show that there is a family of polynomials {Pn}, where Pn is a polynomial in n variables of degree at most d = O(log2 n), such that • Pn can be computed by linear sized homogeneous depth-5 circuits. • Pn can be computed by poly(n) sized non-homogeneous depth-3 circuits. • Any homogeneous depth-4 circuit computing Pn must have size at least nΩ( √ d). This shows that the parameters for the depth reduction results of [AV08, Koi12, Tav15] are tight for extremely restricted classes of arithmetic circuits, for instance homogeneous depth-5 circuits and non-homogeneous depth-3 circuits, and over an appropriate range of parameters, qualitatively improve a result of Kumar and Saraf [KS14b], which showed that the parameters of depth reductions are optimal for algebraic branching programs.
منابع مشابه
Boolean complexity classes vs. their arithmetic analogs
This paper provides logspace and small circuit depth analogs of the result of Valiant and Vazirani, which is a randomized (or nonuniform) reduction from NP to its arithmetic analog ⊕P . We show a similar randomized reduction between the Boolean classes NL and semiunbounded fan-in Boolean circuits and their arithmetic counterparts. These reductions are based on the Isolation Lemma of Mulmuley, V...
متن کاملSome Lower Bound Results for Set-Multilinear Arithmetic Computations
In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to show lower bound results. Some of our results extend existing lower bounds, while others are new and raise open questions. Specifically, our main results are t...
متن کاملConvolutional Rectifier Networks as Generalized Tensor Decompositions
Convolutional rectifier networks, i.e. convolutional neural networks with rectified linear activation and max or average pooling, are the cornerstone of modern deep learning. However, despite their wide use and success, our theoretical understanding of the expressive properties that drive these networks is partial at best. On other hand, we have a much firmer grasp of these issues in the world ...
متن کاملA High-Speed Dual-Bit Parallel Adder based on Carbon Nanotube FET technology for use in arithmetic units
In this paper, a Dual-Bit Parallel Adder (DBPA) based on minority function using Carbon-Nanotube Field-Effect Transistor (CNFET) is proposed. The possibility of having several threshold voltage (Vt) levels by CNFETs leading to wide use of them in designing of digital circuits. The main goal of designing proposed DBPA is to reduce critical path delay in adder circuits. The proposed design positi...
متن کاملOn the Expressive Power of Deep Learning: A Tensor Analysis
It has long been conjectured that hypothesis spaces suitable for data that is compositional in nature, such as text or images, may be more efficiently represented with deep hierarchical architectures than with shallow ones. Despite the vast empirical evidence, formal arguments to date are limited and do not capture the kind of networks used in practice. Using tensor factorization, we derive a u...
متن کامل