Reducing Testing Affine Spaces to Testing Linearity of Functions

Abstract

For any finite field $\mathcal{F}$ and $k<\ell$, we consider the task of testing whether a function $f:\mathcal{F}^\ell \rightarrow \{0,1\}$ is the indicator function of an $(\ell -k)$-dimensional affine space. For the case of $\mathcal{F}=\mathrm{GF}(2)$, an optimal tester for this property was presented by Parnas, Ron, and Samorodnitsky (SIDMA 2002), by mimicking the celebrated linearity tester of Blum, Luby and Rubinfeld (JCSS 1993) and its analysis. We show that the former task (i.e., testing $(\ell -k)$-dimensional affine spaces) can be efficiently reduced to testing the linearity of a related function $g:\mathcal{F}^\ell \rightarrow \mathcal{F}^k$. This reduction yields an almost optimal tester for affine spaces (represented by their indicator function).

Recalling that Parnas, Ron, and Samorodnitsky used testing $(\ell -k)$-dimensional affine spaces as the first step in a two-step procedure for testing k-monomials, we also show that the second step in their procedure can be reduced to testing whether the foregoing function g depends on k of its variables.

Appendix: On Testing k-Linearity

For an arbitrary finite field $\mathcal{F}$ and integers $k,\ell ,m\in \mathbb N$, we say that $f:\mathcal{F}^\ell \rightarrow \mathcal{F}^m$ is k-linear if f is a linear function that depends on at most k (out of its $\ell$) variables. Assuming that the tested function f is linear, we present a simple test that always accepts k-linear functions and rejects (w.h.p.) linear functions that are not k-linear.

The tester uses the key idea of the k-junta test of Fischer et al. [5], which is to randomly partition the $\ell$ variables to $O(k^2)$ sets, hoping that (in the case of a yes-instance) each set contains at most one variable that influences the function’s value, and detecting which of the sets contain such a variable, hereafter called an influential variable. We accept if and only if at most k sets are detected as containing influential variables. To check whether a set $S\subset [\ell ]$ contains influential variables, we select an arbitrary (e.g., random) assignment to the variables in ${\overline{S}}=[\ell ]\setminus S$ and two random (and independent) assignments to the variables in S, and check whether the function’s value is the same under both assignments.¹²

Given that the tested function is linear, the analysis of this procedure is quite straightforward (and, indeed, much simpler than in [5]): If S contains some influential variables of f, then the function’s value changes with probability at least $1-|\mathcal{F}|^{-1}$ (e.g., it is exactly $1-|\mathcal{F}|^{-1}$ in case $m=1$), whereas the value does not change if S contains no influential variable. The complexity of this procedure is $\varOmega (k^2)$, but we can improve it by using two levels of partitions. Specifically, we suggest the following procedure (where the stated expectations refer to the case that the function is O(k)-linear).¹³

• 1. Select a random partition of $[\ell ]$ into $t = k/\log k$ parts, expecting $O(\log k)$ influential variables in each part.

• 2. For each part S in this t-partition, perform the following (three-step) trial for $t'=O(\log k)$ times.

  • (a) Select a random partition of S to $O(\log k)^2$ sub-parts, expecting at most a single influential variable in each sub-part.

  • (b) For each sub-part, check whether it contains influential variables (by selecting two random assignments as described above). Actually, we perform this check $O(\log k)$ times such that a sub-part that contains influential variables is detected to be so with probability at least $1-o(1/k)$.

  • (c) Set the vote of the current trial to equal the number of sub-parts that were detected as containing influential variables.

    (Note that if the sub-partition selected in Step (a) is good (i.e., each sub-part contains at most one influential variable) and all relevant sub-parts were detected as containing influential variables, then the current vote equals the number of influential variables in S.)

  For each S, set the verdict for number of influential variables in S to equal the largest vote obtained in the $t'$ corresponding trials, expecting the answer to be correct with probability $1-o(1/t)$.

• 3. Accept if and only if the sum of the verdicts is at most k.

The foregoing test has complexity $t \cdot t'\cdot O(\log k)^3={\widetilde{O}}(k)$, but this can be improved using an adaptive procedure. Specifically, the following recursive procedure was suggested to us by Eric Blais.

• 1. Select a random partition of $[\ell ]$ into k parts, expecting $\varOmega (k)$ parts with a single variable.

• 2. For each part S in this k-partition, determine whether it contains exactly one variable by performing the following (two-step) trial for $t'=O(\log k)$ times.

  • (a) Select a random 2-partition of S and check whether each sub-part contains influential variables. (We stress that each of these two checks is performed once.)

  • (b) Set the vote of the current trial to equal the number of sub-parts that were detected as containing influential variables.

  For each S, declare this part as having a single influential variable if and only if the maximum vote for it equals 1.

  (Note that, with probability $1-o(1/k)$, the declaration regarding S is correct; that is, S is declared as having a single influential variable if and only if this is actually the case.)

• 3. Let U be the union of the parts declared to have a single influential variable, and $k'$ be their number. If more than k parts were found to contain influential variables (e.g., $k'>k$) then reject; if $k'\le k$ and no part was found to contain more than one influential variable then accept. Otherwise, recurse with the function restricted to $[\ell ]\setminus U$ and $k\leftarrow k-k'$.

Observing that for k-linear functions, with high probability, the number of parts in all recursion calls is O(k), it follows that this procedure has complexity $O(k\log k)$.