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- W2952402601 abstract "The degree-$d$ Chow parameters of a Boolean function $f: {-1,1}^n to mathbb{R}$ are its degree at most $d$ Fourier coefficients. It is well-known that degree-$d$ Chow parameters uniquely characterize degree-$d$ polynomial threshold functions (PTFs) within the space of all bounded functions. In this paper, we prove a robust version of this theorem: For $f$ any Boolean degree-$d$ PTF and $g$ any bounded function, if the degree-$d$ Chow parameters of $f$ are close to the degree-$d$ Chow parameters of $g$ in $ell_2$-norm, then $f$ is close to $g$ in $ell_1$-distance. Notably, our bound relating the two distances is completely independent of the dimension $n$. That is, we show that Boolean degree-$d$ PTFs are {em robustly identifiable} from their degree-$d$ Chow parameters. Results of this form had been shown for the $d=1$ case~cite{OS11:chow, DeDFS14}, but no non-trivial bound was previously known for $d >1$. Our robust identifiability result gives the following algorithmic applications: First, we show that Boolean degree-$d$ PTFs can be efficiently approximately reconstructed from approximations to their degree-$d$ Chow parameters. This immediately implies that degree-$d$ PTFs are efficiently learnable in the uniform distribution $d$-RFA model~cite{BenDavidDichterman:98}. As a byproduct of our approach, we also obtain the first low integer-weight approximations of degree-$d$ PTFs, for $d>1$. As our second application, our robust identifiability result gives the first efficient algorithm, with dimension-independent error guarantees, for malicious learning of Boolean degree-$d$ PTFs under the uniform distribution." @default.
- W2952402601 created "2019-06-27" @default.
- W2952402601 creator A5035962853 @default.
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- W2952402601 date "2018-01-01" @default.
- W2952402601 modified "2023-09-24" @default.
- W2952402601 title "Degree-$d$ Chow Parameters Robustly Determine Degree-$d$ PTFs (and Algorithmic Applications)." @default.
- W2952402601 hasPublicationYear "2018" @default.
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