13,956 research outputs found
Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces
The \emph{Chow parameters} of a Boolean function
are its degree-0 and degree-1 Fourier coefficients. It has been known
since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of
any linear threshold function uniquely specify within the space of all
Boolean functions, but until recently (O'Donnell and Servedio) nothing was
known about efficient algorithms for \emph{reconstructing} (exactly or
approximately) from exact or approximate values of its Chow parameters. We
refer to this reconstruction problem as the \emph{Chow Parameters Problem.}
Our main result is a new algorithm for the Chow Parameters Problem which,
given (sufficiently accurate approximations to) the Chow parameters of any
linear threshold function , runs in time \tilde{O}(n^2)\cdot
(1/\eps)^{O(\log^2(1/\eps))} and with high probability outputs a
representation of an LTF that is \eps-close to . The only previous
algorithm (O'Donnell and Servedio) had running time \poly(n) \cdot
2^{2^{\tilde{O}(1/\eps^2)}}.
As a byproduct of our approach, we show that for any linear threshold
function over , there is a linear threshold function which
is \eps-close to and has all weights that are integers at most \sqrt{n}
\cdot (1/\eps)^{O(\log^2(1/\eps))}. This significantly improves the best
previous result of Diakonikolas and Servedio which gave a \poly(n) \cdot
2^{\tilde{O}(1/\eps^{2/3})} weight bound, and is close to the known lower
bound of (1/\eps)^{\Omega(\log \log (1/\eps))}\} (Goldberg,
Servedio). Our techniques also yield improved algorithms for related problems
in learning theory
Learning Geometric Concepts with Nasty Noise
We study the efficient learnability of geometric concept classes -
specifically, low-degree polynomial threshold functions (PTFs) and
intersections of halfspaces - when a fraction of the data is adversarially
corrupted. We give the first polynomial-time PAC learning algorithms for these
concept classes with dimension-independent error guarantees in the presence of
nasty noise under the Gaussian distribution. In the nasty noise model, an
omniscient adversary can arbitrarily corrupt a small fraction of both the
unlabeled data points and their labels. This model generalizes well-studied
noise models, including the malicious noise model and the agnostic (adversarial
label noise) model. Prior to our work, the only concept class for which
efficient malicious learning algorithms were known was the class of
origin-centered halfspaces.
Specifically, our robust learning algorithm for low-degree PTFs succeeds
under a number of tame distributions -- including the Gaussian distribution
and, more generally, any log-concave distribution with (approximately) known
low-degree moments. For LTFs under the Gaussian distribution, we give a
polynomial-time algorithm that achieves error , where
is the noise rate. At the core of our PAC learning results is an efficient
algorithm to approximate the low-degree Chow-parameters of any bounded function
in the presence of nasty noise. To achieve this, we employ an iterative
spectral method for outlier detection and removal, inspired by recent work in
robust unsupervised learning. Our aforementioned algorithm succeeds for a range
of distributions satisfying mild concentration bounds and moment assumptions.
The correctness of our robust learning algorithm for intersections of
halfspaces makes essential use of a novel robust inverse independence lemma
that may be of broader interest
Comparing and combining measurement-based and driven-dissipative entanglement stabilization
We demonstrate and contrast two approaches to the stabilization of qubit
entanglement by feedback. Our demonstration is built on a feedback platform
consisting of two superconducting qubits coupled to a cavity which are measured
by a nearly-quantum-limited measurement chain and controlled by high-speed
classical logic circuits. This platform is used to stabilize entanglement by
two nominally distinct schemes: a "passive" reservoir engineering method and an
"active" correction based on conditional parity measurements. In view of the
instrumental roles that these two feedback paradigms play in quantum
error-correction and quantum control, we directly compare them on the same
experimental setup. Further, we show that a second layer of feedback can be
added to each of these schemes, which heralds the presence of a high-fidelity
entangled state in realtime. This "nested" feedback brings about a marked
entanglement fidelity improvement without sacrificing success probability.Comment: 40 pages, 12 figure
The anatomy of the Gunn laser
A monopolar GaAs Fabry–Pérot cavity laser based on the Gunn effect is studied both experimentally and theoretically. The light emission occurs via the band-to-band recombination of impact-ionized excess carriers in the propagating space-charge (Gunn) domains. Electroluminescence spectrum from the cleaved end-facet emission of devices with Ga1−xAlxAs (x = 0.32) waveguides shows clearly a preferential mode at a wavelength around 840 nm at T = 95 K. The threshold laser gain is assessed by using an impact ionization coefficient resulting from excess carriers inside the high-field domain
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