11,738 research outputs found
Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits
In order to formally understand the power of neural computing, we first need
to crack the frontier of threshold circuits with two and three layers, a regime
that has been surprisingly intractable to analyze. We prove the first
super-linear gate lower bounds and the first super-quadratic wire lower bounds
for depth-two linear threshold circuits with arbitrary weights, and depth-three
majority circuits computing an explicit function.
We prove that for all , the
linear-time computable Andreev's function cannot be computed on a
-fraction of -bit inputs by depth-two linear threshold
circuits of gates, nor can it be computed with
wires. This establishes an average-case
``size hierarchy'' for threshold circuits, as Andreev's function is computable
by uniform depth-two circuits of linear threshold gates, and by
uniform depth-three circuits of majority gates.
We present a new function in based on small-biased sets, which
we prove cannot be computed by a majority vote of depth-two linear threshold
circuits with gates, nor with
wires.
We give tight average-case (gate and wire) complexity results for
computing PARITY with depth-two threshold circuits; the answer turns out to be
the same as for depth-two majority circuits.
The key is a new random restriction lemma for linear threshold functions. Our
main analytical tool is the Littlewood-Offord Lemma from additive
combinatorics
Small Depth Quantum Circuits
Small depth quantum circuits have proved to be unexpectedly powerful in comparison to their classical counterparts. We survey some of the recent work on this and present some open problems.National Security Agency; Advanced Research and Development Agency under Army Research Office (DAAD 19-02-1-0058
Polynomials that Sign Represent Parity and Descartes' Rule of Signs
A real polynomial sign represents if
for every , the sign of equals
. Such sign representations are well-studied in computer
science and have applications to computational complexity and computational
learning theory. In this work, we present a systematic study of tradeoffs
between degree and sparsity of sign representations through the lens of the
parity function. We attempt to prove bounds that hold for any choice of set
. We show that sign representing parity over with the
degree in each variable at most requires sparsity at least . We show
that a tradeoff exists between sparsity and degree, by exhibiting a sign
representation that has higher degree but lower sparsity. We show a lower bound
of on the sparsity of polynomials of any degree representing
parity over . We prove exact bounds on the sparsity of such
polynomials for any two element subset . The main tool used is Descartes'
Rule of Signs, a classical result in algebra, relating the sparsity of a
polynomial to its number of real roots. As an application, we use bounds on
sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with
a Threshold Gate at the top. We use this to give a simple proof that such
circuits need size to compute parity, which improves the previous bound
of due to Goldmann (1997). We show a tight lower bound of
for the inner product function over .Comment: To appear in Computational Complexit
Programmable neural logic
Circuits of threshold elements (Boolean input, Boolean output neurons) have been shown to be surprisingly powerful. Useful functions such as XOR, ADD and MULTIPLY can be implemented by such circuits more efficiently than by traditional AND/OR circuits. In view of that, we have designed and built a programmable threshold element. The weights are stored on polysilicon floating gates, providing long-term retention without refresh. The weight value is increased using tunneling and decreased via hot electron injection. A weight is stored on a single transistor allowing the development of dense arrays of threshold elements. A 16-input programmable neuron was fabricated in the standard 2 μm double-poly, analog process available from MOSIS.
We also designed and fabricated the multiple threshold element introduced in [5]. It presents the advantage of reducing the area of the layout from O(n^2) to O(n); (n being the number of variables) for a broad class of Boolean functions, in particular symmetric Boolean functions such as PARITY.
A long term goal of this research is to incorporate programmable single/multiple threshold elements, as building blocks in field programmable gate arrays
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