No existence of linear algorithm for Fourier phase retrieval

Fourier phase retrieval, which seeks to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we give a theoretical understanding of algorithms for Fourier phase retrieval. Particularly, we show if there exists an algorithm which could reconstruct an arbitrary signal ${\mathbf x}\in {\mathbb C}^N$ in $ \mbox{Poly}(N) \log(1/\epsilon)$ time to reach $\epsilon$-precision from its magnitude of discrete Fourier transform and its initial value $x(0)$, then $\mathcal{ P}=\mathcal{NP}$. This demystifies the phenomenon that, although almost all signals are determined uniquely by their Fourier magnitude with a prior conditions, there is no algorithm with theoretical guarantees being proposed over the past few decades. Our proofs employ the result in computational complexity theory that Product Partition problem is NP-complete in the strong sense.

"Let Your Characters Tell Their Story": A Dataset for Character-Centric Narrative Understanding

When reading a literary piece, readers often make inferences about various characters' roles, personalities, relationships, intents, actions, etc. While humans can readily draw upon their past experiences to build such a character-centric view of the narrative, understanding characters in narratives can be a challenging task for machines. To encourage research in this field of character-centric narrative understanding, we present LiSCU -- a new dataset of literary pieces and their summaries paired with descriptions of characters that appear in them. We also introduce two new tasks on LiSCU: Character Identification and Character Description Generation. Our experiments with several pre-trained language models adapted for these tasks demonstrate that there is a need for better models of narrative comprehension.