20,032 research outputs found
Exponential Separation of Quantum Communication and Classical Information
We exhibit a Boolean function for which the quantum communication complexity
is exponentially larger than the classical information complexity. An
exponential separation in the other direction was already known from the work
of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that
these two complexity measures are incomparable. As classical information
complexity is an upper bound on quantum information complexity, which in turn
is equal to amortized quantum communication complexity, our work implies that a
tight direct sum result for distributional quantum communication complexity
cannot hold. The function we use to present such a separation is the Symmetric
k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057],
whose classical communication complexity is exponentially larger than its
classical information complexity. In this paper, we show that the quantum
communication complexity of this function is polynomially equivalent to its
classical communication complexity. The high-level idea behind our proof is
arguably the simplest so far for such an exponential separation between
information and communication, driven by a sequence of round-elimination
arguments, allowing us to simplify further the approach of Rao and Sinha.
As another application of the techniques that we develop, we give a simple
proof for an optimal trade-off between Alice's and Bob's communication while
computing the related Greater-Than function on n bits: say Bob communicates at
most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when
allowing pre-shared entanglement. We also present a classical protocol
achieving this bound.Comment: v1, 36 pages, 3 figure
Privacy-Preserving Shortest Path Computation
Navigation is one of the most popular cloud computing services. But in
virtually all cloud-based navigation systems, the client must reveal her
location and destination to the cloud service provider in order to learn the
fastest route. In this work, we present a cryptographic protocol for navigation
on city streets that provides privacy for both the client's location and the
service provider's routing data. Our key ingredient is a novel method for
compressing the next-hop routing matrices in networks such as city street maps.
Applying our compression method to the map of Los Angeles, for example, we
achieve over tenfold reduction in the representation size. In conjunction with
other cryptographic techniques, this compressed representation results in an
efficient protocol suitable for fully-private real-time navigation on city
streets. We demonstrate the practicality of our protocol by benchmarking it on
real street map data for major cities such as San Francisco and Washington,
D.C.Comment: Extended version of NDSS 2016 pape
What Can We Learn Privately?
Learning problems form an important category of computational tasks that
generalizes many of the computations researchers apply to large real-life data
sets. We ask: what concept classes can be learned privately, namely, by an
algorithm whose output does not depend too heavily on any one input or specific
training example? More precisely, we investigate learning algorithms that
satisfy differential privacy, a notion that provides strong confidentiality
guarantees in contexts where aggregate information is released about a database
containing sensitive information about individuals. We demonstrate that,
ignoring computational constraints, it is possible to privately agnostically
learn any concept class using a sample size approximately logarithmic in the
cardinality of the concept class. Therefore, almost anything learnable is
learnable privately: specifically, if a concept class is learnable by a
(non-private) algorithm with polynomial sample complexity and output size, then
it can be learned privately using a polynomial number of samples. We also
present a computationally efficient private PAC learner for the class of parity
functions. Local (or randomized response) algorithms are a practical class of
private algorithms that have received extensive investigation. We provide a
precise characterization of local private learning algorithms. We show that a
concept class is learnable by a local algorithm if and only if it is learnable
in the statistical query (SQ) model. Finally, we present a separation between
the power of interactive and noninteractive local learning algorithms.Comment: 35 pages, 2 figure
An operad for splicing
A new topological operad is introduced, called the splicing operad. This
operad acts on a broad class of spaces of self-embeddings N --> N where N is a
manifold. The action of this operad on EC(j,M) (self embeddings R^j x M --> R^j
x M with support in I^j x M) is an extension of the action of the operad of
(j+1)-cubes on this space. Moreover the action of the splicing operad encodes
Larry Siebenmann's splicing construction for knots in S^3 in the j=1, M=D^2
case. The space of long knots in R^3 (denoted K_{3,1}) was shown to be a free
2-cubes object with free generating subspace P, the subspace of long knots that
are prime with respect to the connect-sum operation. One of the main results of
this paper is that K_{3,1} is free with respect to the splicing operad action,
but the free generating space is the much `smaller' space of torus and
hyperbolic knots TH \subset K_{3,1}. Moreover, the splicing operad for K_{3,1}
has a `simple' homotopy-type as an operad.Comment: 34 pages, 16 diagrams. V3->V4: Explicit definition of sigma^* \wr
G-operads given, and proof included splicing operad is such. Re-wrote proof
of Theorem 5.13 to make the equivariance of the maps more explicit. Cut down
on extraneous notation. Fixed references, some small typo
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