5,115 research outputs found
Analysis of classifiers' robustness to adversarial perturbations
The goal of this paper is to analyze an intriguing phenomenon recently
discovered in deep networks, namely their instability to adversarial
perturbations (Szegedy et. al., 2014). We provide a theoretical framework for
analyzing the robustness of classifiers to adversarial perturbations, and show
fundamental upper bounds on the robustness of classifiers. Specifically, we
establish a general upper bound on the robustness of classifiers to adversarial
perturbations, and then illustrate the obtained upper bound on the families of
linear and quadratic classifiers. In both cases, our upper bound depends on a
distinguishability measure that captures the notion of difficulty of the
classification task. Our results for both classes imply that in tasks involving
small distinguishability, no classifier in the considered set will be robust to
adversarial perturbations, even if a good accuracy is achieved. Our theoretical
framework moreover suggests that the phenomenon of adversarial instability is
due to the low flexibility of classifiers, compared to the difficulty of the
classification task (captured by the distinguishability). Moreover, we show the
existence of a clear distinction between the robustness of a classifier to
random noise and its robustness to adversarial perturbations. Specifically, the
former is shown to be larger than the latter by a factor that is proportional
to \sqrt{d} (with d being the signal dimension) for linear classifiers. This
result gives a theoretical explanation for the discrepancy between the two
robustness properties in high dimensional problems, which was empirically
observed in the context of neural networks. To the best of our knowledge, our
results provide the first theoretical work that addresses the phenomenon of
adversarial instability recently observed for deep networks. Our analysis is
complemented by experimental results on controlled and real-world data
On polyhedral approximations of the positive semidefinite cone
Let be the set of positive semidefinite matrices of trace
equal to one, also known as the set of density matrices. We prove two results
on the hardness of approximating with polytopes. First, we show that if and is an arbitrary matrix of trace equal to one, any
polytope such that must have
linear programming extension complexity at least where is a constant that depends on . Second, we show that any polytope
such that and such that the Gaussian width of is at most
twice the Gaussian width of must have extension complexity at least
. The main ingredient of our proofs is hypercontractivity of
the noise operator on the hypercube.Comment: 12 page
Algorithmic Aspects of Optimal Channel Coding
A central question in information theory is to determine the maximum success
probability that can be achieved in sending a fixed number of messages over a
noisy channel. This was first studied in the pioneering work of Shannon who
established a simple expression characterizing this quantity in the limit of
multiple independent uses of the channel. Here we consider the general setting
with only one use of the channel. We observe that the maximum success
probability can be expressed as the maximum value of a submodular function.
Using this connection, we establish the following results:
1. There is a simple greedy polynomial-time algorithm that computes a code
achieving a (1-1/e)-approximation of the maximum success probability. Moreover,
for this problem it is NP-hard to obtain an approximation ratio strictly better
than (1-1/e).
2. Shared quantum entanglement between the sender and the receiver can
increase the success probability by a factor of at most 1/(1-1/e). In addition,
this factor is tight if one allows an arbitrary non-signaling box between the
sender and the receiver.
3. We give tight bounds on the one-shot performance of the meta-converse of
Polyanskiy-Poor-Verdu.Comment: v2: 16 pages. Added alternate proof of main result with random codin
An ontology-based approach to relax traffic regulation for autonomous vehicle assistance
Traffic regulation must be respected by all vehicles, either human- or
computer- driven. However, extreme traffic situations might exhibit practical
cases in which a vehicle should safely and reasonably relax traffic regulation,
e.g., in order not to be indefinitely blocked and to keep circulating. In this
paper, we propose a high-level representation of an automated vehicle, other
vehicles and their environment, which can assist drivers in taking such
"illegal" but practical relaxation decisions. This high-level representation
(an ontology) includes topological knowledge and inference rules, in order to
compute the next high-level motion an automated vehicle should take, as
assistance to a driver. Results on practical cases are presented
On simultaneous min-entropy smoothing
In the context of network information theory, one often needs a multiparty
probability distribution to be typical in several ways simultaneously. When
considering quantum states instead of classical ones, it is in general
difficult to prove the existence of a state that is jointly typical. Such a
difficulty was recently emphasized and conjectures on the existence of such
states were formulated. In this paper, we consider a one-shot multiparty
typicality conjecture. The question can then be stated easily: is it possible
to smooth the largest eigenvalues of all the marginals of a multipartite state
{\rho} simultaneously while staying close to {\rho}? We prove the answer is yes
whenever the marginals of the state commute. In the general quantum case, we
prove that simultaneous smoothing is possible if the number of parties is two
or more generally if the marginals to optimize satisfy some non-overlap
property.Comment: 5 page
Scrambling speed of random quantum circuits
Random transformations are typically good at "scrambling" information.
Specifically, in the quantum setting, scrambling usually refers to the process
of mapping most initial pure product states under a unitary transformation to
states which are macroscopically entangled, in the sense of being close to
completely mixed on most subsystems containing a fraction fn of all n particles
for some constant f. While the term scrambling is used in the context of the
black hole information paradox, scrambling is related to problems involving
decoupling in general, and to the question of how large isolated many-body
systems reach local thermal equilibrium under their own unitary dynamics.
Here, we study the speed at which various notions of scrambling/decoupling
occur in a simplified but natural model of random two-particle interactions:
random quantum circuits. For a circuit representing the dynamics generated by a
local Hamiltonian, the depth of the circuit corresponds to time. Thus, we
consider the depth of these circuits and we are typically interested in what
can be done in a depth that is sublinear or even logarithmic in the size of the
system. We resolve an outstanding conjecture raised in the context of the black
hole information paradox with respect to the depth at which a typical quantum
circuit generates an entanglement assisted encoding against the erasure
channel. In addition, we prove that typical quantum circuits of poly(log n)
depth satisfy a stronger notion of scrambling and can be used to encode alpha n
qubits into n qubits so that up to beta n errors can be corrected, for some
constants alpha, beta > 0.Comment: 24 pages, 2 figures. Superseded by http://arxiv.org/abs/1307.063
Decoupling with random quantum circuits
Decoupling has become a central concept in quantum information theory with
applications including proving coding theorems, randomness extraction and the
study of conditions for reaching thermal equilibrium. However, our
understanding of the dynamics that lead to decoupling is limited. In fact, the
only families of transformations that are known to lead to decoupling are
(approximate) unitary two-designs, i.e., measures over the unitary group which
behave like the Haar measure as far as the first two moments are concerned.
Such families include for example random quantum circuits with O(n^2) gates,
where n is the number of qubits in the system under consideration. In fact, all
known constructions of decoupling circuits use \Omega(n^2) gates.
Here, we prove that random quantum circuits with O(n log^2 n) gates satisfy
an essentially optimal decoupling theorem. In addition, these circuits can be
implemented in depth O(log^3 n). This proves that decoupling can happen in a
time that scales polylogarithmically in the number of particles in the system,
provided all the particles are allowed to interact. Our proof does not proceed
by showing that such circuits are approximate two-designs in the usual sense,
but rather we directly analyze the decoupling property.Comment: 25 page
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