268 research outputs found
The Role of Riemannian Manifolds in Computer Vision: From Coding to Deep Metric Learning
A diverse number of tasks in computer vision and machine learning
enjoy from representations of data that are compact yet
discriminative, informative and robust to critical measurements.
Two notable representations are offered by Region Covariance
Descriptors (RCovD) and linear subspaces which are naturally
analyzed through the manifold of Symmetric Positive Definite
(SPD) matrices and the Grassmann manifold, respectively, two
widely used types of Riemannian manifolds in computer vision.
As our first objective, we examine image and video-based
recognition applications where the local descriptors have the
aforementioned Riemannian structures, namely the SPD or linear
subspace structure. Initially, we provide a solution to compute
Riemannian version of the conventional Vector of Locally
aggregated Descriptors (VLAD), using geodesic distance of the
underlying manifold as the nearness measure. Next, by having a
closer look at the resulting codes, we formulate a new concept
which we name Local Difference Vectors (LDV). LDVs enable us to
elegantly expand our Riemannian coding techniques to any
arbitrary metric as well as provide intrinsic solutions to
Riemannian sparse coding and its variants when local structured
descriptors are considered.
We then turn our attention to two special types of covariance
descriptors namely infinite-dimensional RCovDs and rank-deficient
covariance matrices for which the underlying Riemannian
structure, i.e. the manifold of SPD matrices is out of reach to
great extent. %Generally speaking, infinite-dimensional RCovDs
offer better discriminatory power over their low-dimensional
counterparts.
To overcome this difficulty, we propose to approximate the
infinite-dimensional RCovDs by making use of two feature
mappings, namely random Fourier features and the Nystrom method.
As for the rank-deficient covariance matrices, unlike most
existing approaches that employ inference tools by predefined
regularizers, we derive positive definite kernels that can be
decomposed into the kernels on the cone of SPD matrices and
kernels on the Grassmann manifolds and show their effectiveness
for image set classification task.
Furthermore, inspired by attractive properties of Riemannian
optimization techniques, we extend the recently introduced Keep
It Simple and Straightforward MEtric learning (KISSME) method to
the scenarios where input data is non-linearly distributed. To
this end, we make use of the infinite dimensional covariance
matrices and propose techniques towards projecting on the
positive cone in a Reproducing Kernel Hilbert Space (RKHS).
We also address the sensitivity issue of the KISSME to the input
dimensionality. The KISSME algorithm is greatly dependent on
Principal Component Analysis (PCA) as a preprocessing step which
can lead to difficulties, especially when the dimensionality is
not meticulously set.
To address this issue, based on the KISSME algorithm, we develop
a Riemannian framework to jointly learn a mapping performing
dimensionality reduction and a metric in the induced space.
Lastly, in line with the recent trend in metric learning, we
devise end-to-end learning of a generic deep network for metric
learning using our derivation
Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis
A method for synthesizing quantum gates is presented based on interpolation
methods applied to operators in Hilbert space. Starting from the diagonal forms
of specific generating seed operators with non-degenerate eigenvalue spectrum
one obtains for arity-one a complete family of logical operators corresponding
to all the one-argument logical connectives. Scaling-up to n-arity gates is
obtained by using the Kronecker product and unitary transformations. The
quantum version of the Fourier transform of Boolean functions is presented and
a Reed-Muller decomposition for quantum logical gates is derived. The common
control gates can be easily obtained by considering the logical correspondence
between the control logic operator and the binary propositional logic operator.
A new polynomial and exponential formulation of the Toffoli gate is presented.
The method has parallels to quantum gate-T optimization methods using powers of
multilinear operator polynomials. The method is then applied naturally to
alphabets greater than two for multi-valued logical gates used for quantum
Fourier transform, min-max decision circuits and multivalued adders
A topological approximation of the nonlinear Anderson model
We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review
Quantum Kaleidoscopes and Bell's theorem
A quantum kaleidoscope is defined as a set of observables, or states,
consisting of many different subsets that provide closely related proofs of the
Bell-Kochen-Specker (BKS) and Bell nonlocality theorems. The kaleidoscopes
prove the BKS theorem through a simple parity argument, which also doubles as a
proof of Bell's nonlocality theorem if use is made of the right sort of
entanglement. Three closely related kaleidoscopes are introduced and discussed
in this paper: a 15-observable kaleidoscope, a 24-state kaleidoscope and a
60-state kaleidoscope. The close relationship of these kaleidoscopes to a
configuration of 12 points and 16 lines known as Reye's configuration is
pointed out. The "rotations" needed to make each kaleidoscope yield all its
apparitions are laid out. The 60-state kaleidoscope, whose underlying
geometrical structure is that of ten interlinked Reye's configurations
(together with their duals), possesses a total of 1120 apparitions that provide
proofs of the two Bell theorems. Some applications of these kaleidoscopes to
problems in quantum tomography and quantum state estimation are discussed.Comment: Two new references (No. 21 and 22) to related work have been adde
Statistically Motivated Second Order Pooling
Second-order pooling, a.k.a.~bilinear pooling, has proven effective for deep
learning based visual recognition. However, the resulting second-order networks
yield a final representation that is orders of magnitude larger than that of
standard, first-order ones, making them memory-intensive and cumbersome to
deploy. Here, we introduce a general, parametric compression strategy that can
produce more compact representations than existing compression techniques, yet
outperform both compressed and uncompressed second-order models. Our approach
is motivated by a statistical analysis of the network's activations, relying on
operations that lead to a Gaussian-distributed final representation, as
inherently used by first-order deep networks. As evidenced by our experiments,
this lets us outperform the state-of-the-art first-order and second-order
models on several benchmark recognition datasets.Comment: Accepted to ECCV 2018. Camera ready version. 14 page, 5 figures, 3
table
- …