31 research outputs found

### Near-optimal Bootstrapping of Hitting Sets for Algebraic Models

The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel
[Ore22,DL78,Zip79,Sch80] states that any nonzero polynomial $f(x_1,\ldots,
x_n)$ of degree at most $s$ will evaluate to a nonzero value at some point on a
grid $S^n \subseteq \mathbb{F}^n$ with $|S| > s$. Thus, there is an explicit
hitting set for all $n$-variate degree $s$, size $s$ algebraic circuits of size
$(s+1)^n$.
In this paper, we prove the following results:
- Let $\epsilon > 0$ be a constant. For a sufficiently large constant $n$ and
all $s > n$, if we have an explicit hitting set of size $(s+1)^{n-\epsilon}$
for the class of $n$-variate degree $s$ polynomials that are computable by
algebraic circuits of size $s$, then for all $s$, we have an explicit hitting
set of size $s^{\exp \circ \exp (O(\log^\ast s))}$ for $s$-variate circuits of
degree $s$ and size $s$. That is, if we can obtain a barely non-trivial
exponent compared to the trivial $(s+1)^{n}$ sized hitting set even for
constant variate circuits, we can get an almost complete derandomization of
PIT.
- The above result holds when "circuits" are replaced by "formulas" or
"algebraic branching programs".
This extends a recent surprising result of Agrawal, Ghosh and Saxena [AGS18]
who proved the same conclusion for the class of algebraic circuits, if the
hypothesis provided a hitting set of size at most $(s^{n^{0.5 - \delta}})$
(where $\delta>0$ is any constant). Hence, our work significantly weakens the
hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial
saving over the trivial hitting set, and also presents the first such result
for algebraic branching programs and formulas.Comment: The main result has been strengthened significantly, compared to the
older version of the paper. Additionally, the stronger theorem now holds even
for subclasses of algebraic circuits, such as algebraic formulas and
algebraic branching program

### Constructing Faithful Homomorphisms over Fields of Finite Characteristic

We study the question of algebraic rank or transcendence degree preserving
homomorphisms over finite fields. This concept was first introduced by Beecken,
Mittmann and Saxena (Information and Computing, 2013), and exploited by them,
and Agrawal, Saha, Saptharishi and Saxena (Journal of Computing, 2016) to
design algebraic independence based identity tests using the Jacobian criterion
over characteristic zero fields. An analogue of such constructions over finite
characteristic fields was unknown due to the failure of the Jacobian criterion
over finite characteristic fields.
Building on a recent criterion of Pandey, Sinhababu and Saxena (MFCS, 2016),
we construct explicit faithful maps for some natural classes of polynomials in
the positive characteristic field setting, when a certain parameter called the
inseparable degree of the underlying polynomials is bounded (this parameter is
always 1 in fields of characteristic zero). This presents the first
generalisation of some of the results of Beecken et al. and Agrawal et al. in
the positive characteristic setting

### Constructing Faithful Homomorphisms over Fields of Finite Characteristic

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [Malte Beecken et al., 2013] and exploited by them and Agrawal et al. [Manindra Agrawal et al., 2016] to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields were unknown due to the failure of the Jacobian criterion over finite characteristic fields.
Building on a recent criterion of Pandey, Saxena and Sinhababu [Anurag Pandey et al., 2018], we construct explicit faithful maps for some natural classes of polynomials in fields of positive characteristic, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken, Mittmann and Saxena [Malte Beecken et al., 2013] and Agrawal, Saha, Saptharishi, Saxena [Manindra Agrawal et al., 2016] in the positive characteristic setting

### Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

We say that a circuit $C$ over a field $F$ functionally computes an
$n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) =
P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as
formal polynomials. In this paper, we study the question of proving lower
bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for
functional computation. We prove the following results :
1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a
polynomial in $VNP$.
2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits
with bounded individual degree for a polynomial in $VNP$.
Our main motivation for this line of research comes from our observation that
strong enough functional lower bounds for even very special depth-$4$
arithmetic circuits for the Permanent imply a separation between ${\#}P$ and
$ACC$. Thus, improving the second result to get rid of the bounded individual
degree condition could lead to substantial progress in boolean circuit
complexity. Besides, it is known from a recent result of Kumar and Saptharishi
[KS15] that over constant sized finite fields, strong enough average case
functional lower bounds for homogeneous depth-$4$ circuits imply
superpolynomial lower bounds for homogeneous depth-$5$ circuits.
Our proofs are based on a family of new complexity measures called shifted
evaluation dimension, and might be of independent interest

### A note on the elementary HDX construction of Kaufman-Oppenheim

In this note, we give a self-contained and elementary proof of the elementary
construction of spectral high-dimensional expanders using elementary matrices
due to Kaufman and Oppenheim [Proc. 50th ACM Symp. on Theory of Computing
(STOC), 2018]

### An Exponential Lower Bound for Homogeneous Depth-5 Circuits over Finite Fields

In this paper, we show exponential lower bounds for the class of homogeneous depth-5 circuits over all small finite fields. More formally, we show that there is an explicit family {P_d} of polynomials in VNP, where P_d is of degree d in n = d^{O(1)} variables, such that over all finite fields GF(q), any homogeneous depth-5 circuit which computes P_d must have size at least exp(Omega_q(sqrt{d})).
To the best of our knowledge, this is the first super-polynomial lower bound for this class for any non-binary field.
Our proof builds up on the ideas developed on the way to proving lower bounds for homogeneous depth-4 circuits [Gupta et al., Fournier et al., Kayal et al., Kumar-Saraf] and for non-homogeneous depth-3 circuits over finite fields [Grigoriev-Karpinski, Grigoriev-Razborov].
Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from GF(q)^n to GF(q) as opposed to looking at them as a space of formal polynomials and builds over a tighter analysis of the lower bound of Kumar and Saraf [Kumar-Saraf]

### Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

We present a single, common tool to strictly subsume all known cases of
polynomial time blackbox polynomial identity testing (PIT) that have been
hitherto solved using diverse tools and techniques. In particular, we show that
polynomial time hitting-set generators for identity testing of the two
seemingly different and well studied models - depth-3 circuits with bounded top
fanin, and constant-depth constant-read multilinear formulas - can be
constructed using one common algebraic-geometry theme: Jacobian captures
algebraic independence. By exploiting the Jacobian, we design the first
efficient hitting-set generators for broad generalizations of the
above-mentioned models, namely:
(1) depth-3 (Sigma-Pi-Sigma) circuits with constant transcendence degree of
the polynomials computed by the product gates (no bounded top fanin
restriction), and (2) constant-depth constant-occur formulas (no multilinear
restriction).
Constant-occur of a variable, as we define it, is a much more general concept
than constant-read. Also, earlier work on the latter model assumed that the
formula is multilinear. Thus, our work goes further beyond the results obtained
by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et
al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011),
and brings them under one unifying technique.
In addition, using the same Jacobian based approach, we prove exponential
lower bounds for the immanant (which includes permanent and determinant) on the
same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our
results reinforce the intimate connection between identity testing and lower
bounds by exhibiting a concrete mathematical tool - the Jacobian - that is
equally effective in solving both the problems on certain interesting and
previously well-investigated (but not well understood) models of computation

### Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees

We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [Guillaume Lagarde et al., 2016] and Lagarde, Limaye and Srinivasan [Guillaume Lagarde et al., 2017]) and give the following constructions:
- An explicit hitting set of quasipolynomial size for UPT circuits,
- An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes),
- An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant.
The above three results are extensions of the results of [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits.
The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016]