In this paper, we demonstrate that complex bandlimited signals can be uniquely recovered (up to a global phase factor) from two families of wavelet transform magnitudes, sampled at twice the Nyquist rate in two frequency bins. These families come from translations of the fixed dilated Morlet wavelet and a fixed companion wavelet. The companion wavelet admits an augmented Morlet--Gaussian representation. We rewrite the two families as two modulated Gaussian convolution measurements of an auxiliary bandlimited signal. Using a Gaussian convolution uniqueness result, we prove that samples on \((4B)^{-1}\mathbb{Z}\) determine every signal in \(PW_B^2\) (up to a global phase factor). Compared with earlier wavelet phase retrieval results, our approach requires significantly fewer sampling points.
Fan Chang, Joseph Slote, Chun-Kai Tseng, Alexander Volberg, Haonan Zhang
Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture recently solved by OpenAI, \cite{GLsol}. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the second bound \[ \mathbf{BSA}[f]\le \exp(C(d)\sqrt{\log n}). \] This bound was obtained in \cite{FSVZ} without clarifying the magnitude of $C(d)$, which we do below. For lower moments $\bE |\nabla f|^η \le C(d, η)$ there is no dependence on $n$ for $0<η<1$.