1,994 research outputs found
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
Learning from Multi-View Multi-Way Data via Structural Factorization Machines
Real-world relations among entities can often be observed and determined by
different perspectives/views. For example, the decision made by a user on
whether to adopt an item relies on multiple aspects such as the contextual
information of the decision, the item's attributes, the user's profile and the
reviews given by other users. Different views may exhibit multi-way
interactions among entities and provide complementary information. In this
paper, we introduce a multi-tensor-based approach that can preserve the
underlying structure of multi-view data in a generic predictive model.
Specifically, we propose structural factorization machines (SFMs) that learn
the common latent spaces shared by multi-view tensors and automatically adjust
the importance of each view in the predictive model. Furthermore, the
complexity of SFMs is linear in the number of parameters, which make SFMs
suitable to large-scale problems. Extensive experiments on real-world datasets
demonstrate that the proposed SFMs outperform several state-of-the-art methods
in terms of prediction accuracy and computational cost.Comment: 10 page
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