2,512 research outputs found
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
Non-locality and Communication Complexity
Quantum information processing is the emerging field that defines and
realizes computing devices that make use of quantum mechanical principles, like
the superposition principle, entanglement, and interference. In this review we
study the information counterpart of computing. The abstract form of the
distributed computing setting is called communication complexity. It studies
the amount of information, in terms of bits or in our case qubits, that two
spatially separated computing devices need to exchange in order to perform some
computational task. Surprisingly, quantum mechanics can be used to obtain
dramatic advantages for such tasks.
We review the area of quantum communication complexity, and show how it
connects the foundational physics questions regarding non-locality with those
of communication complexity studied in theoretical computer science. The first
examples exhibiting the advantage of the use of qubits in distributed
information-processing tasks were based on non-locality tests. However, by now
the field has produced strong and interesting quantum protocols and algorithms
of its own that demonstrate that entanglement, although it cannot be used to
replace communication, can be used to reduce the communication exponentially.
In turn, these new advances yield a new outlook on the foundations of physics,
and could even yield new proposals for experiments that test the foundations of
physics.Comment: Survey paper, 63 pages LaTeX. A reformatted version will appear in
Reviews of Modern Physic
Hybrid Beamforming via the Kronecker Decomposition for the Millimeter-Wave Massive MIMO Systems
Despite its promising performance gain, the realization of mmWave massive
MIMO still faces several practical challenges. In particular, implementing
massive MIMO in the digital domain requires hundreds of RF chains matching the
number of antennas. Furthermore, designing these components to operate at the
mmWave frequencies is challenging and costly. These motivated the recent
development of hybrid-beamforming where MIMO processing is divided for separate
implementation in the analog and digital domains, called the analog and digital
beamforming, respectively. Analog beamforming using a phase array introduces
uni-modulus constraints on the beamforming coefficients, rendering the
conventional MIMO techniques unsuitable and call for new designs. In this
paper, we present a systematic design framework for hybrid beamforming for
multi-cell multiuser massive MIMO systems over mmWave channels characterized by
sparse propagation paths. The framework relies on the decomposition of analog
beamforming vectors and path observation vectors into Kronecker products of
factors being uni-modulus vectors. Exploiting properties of Kronecker mixed
products, different factors of the analog beamformer are designed for either
nulling interference paths or coherently combining data paths. Furthermore, a
channel estimation scheme is designed for enabling the proposed hybrid
beamforming. The scheme estimates the AoA of data and interference paths by
analog beam scanning and data-path gains by analog beam steering. The
performance of the channel estimation scheme is analyzed. In particular, the
AoA spectrum resulting from beam scanning, which displays the magnitude
distribution of paths over the AoA range, is derived in closed-form. It is
shown that the inter-cell interference level diminishes inversely with the
array size, the square root of pilot sequence length and the spatial separation
between paths.Comment: Submitted to IEEE JSAC Special Issue on Millimeter Wave
Communications for Future Mobile Networks, minor revisio
Strengths and Weaknesses of Quantum Fingerprinting
We study the power of quantum fingerprints in the simultaneous message
passing (SMP) setting of communication complexity. Yao recently showed how to
simulate, with exponential overhead, classical shared-randomness SMP protocols
by means of quantum SMP protocols without shared randomness
(-protocols). Our first result is to extend Yao's simulation to
the strongest possible model: every many-round quantum protocol with unlimited
shared entanglement can be simulated, with exponential overhead, by
-protocols. We apply our technique to obtain an efficient
-protocol for a function which cannot be efficiently solved
through more restricted simulations. Second, we tightly characterize the power
of the quantum fingerprinting technique by making a connection to arrangements
of homogeneous halfspaces with maximal margin. These arrangements have been
well studied in computational learning theory, and we use some strong results
obtained in this area to exhibit weaknesses of quantum fingerprinting. In
particular, this implies that for almost all functions, quantum fingerprinting
protocols are exponentially worse than classical deterministic SMP protocols.Comment: 13 pages, no figures, to appear in CCC'0
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