Spectral Norm of Symmetric Functions

The spectral norm of a Boolean function $f:\{0,1\}^n \to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as $r(f)\log(n/r(f))$ where $r(f) = \max\{r_0,r_1\}$, and $r_0$ and $r_1$ are the smallest integers less than $n/2$ such that $f(x)$ or $f(x) \cdot parity(x)$ is constant for all $x$ with $\sum x_i \in [r_0, n-r_1]$. We mention some applications to the decision tree and communication complexity of symmetric functions

Low-degree tests at large distances

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. In particular, we show that functions with small Gowers uniformity norms behave ``randomly'' with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius

Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers

On the sum of the L1 influences of bounded functions

Let $f\colon \{-1,1\}^n \to [-1,1]$ have degree $d$ as a multilinear polynomial. It is well-known that the total influence of $f$ is at most $d$. Aaronson and Ambainis asked whether the total $L_1$ influence of $f$ can also be bounded as a function of $d$. Ba\v{c}kurs and Bavarian answered this question in the affirmative, providing a bound of $O(d^3)$ for general functions and $O(d^2)$ for homogeneous functions. We improve on their results by providing a bound of $d^2$ for general functions and $O(d\log d)$ for homogeneous functions. In addition, we prove a bound of $d/(2 \pi)+o(d)$ for monotone functions, and provide a matching example.Comment: 16 pages; accepted for publication in the Israel Journal of Mathematic