26 research outputs found
Pooling spaces associated with finite geometry
AbstractMotivated by the works of Ngo and Du [H. Ngo, D. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 55 (2000) 171–182], the notion of pooling spaces was introduced [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Mathematics 282 (2004) 163–169] for a systematic way of constructing pooling designs; note that geometric lattices are among pooling spaces. This paper attempts to draw possible connections from finite geometry and distance regular graphs to pooling spaces: including the projective spaces, the affine spaces, the attenuated spaces, and a few families of geometric lattices associated with the orbits of subspaces under finite classical groups, and associated with d-bounded distance-regular graphs
Applied Harmonic Analysis and Data Processing
Massive data sets have their own architecture. Each data source has an inherent structure, which we should attempt to detect in order to utilize it for applications, such as denoising, clustering, anomaly detection, knowledge extraction, or classification. Harmonic analysis revolves around creating new structures for decomposition, rearrangement and reconstruction of operators and functions—in other words inventing and exploring new architectures for information and inference. Two previous very successful workshops on applied harmonic analysis and sparse approximation have taken place in 2012 and in 2015. This workshop was the an evolution and continuation of these workshops and intended to bring together world leading experts in applied harmonic analysis, data analysis, optimization, statistics, and machine learning to report on recent developments, and to foster new developments and collaborations
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds
Motivated by the vast success of deep convolutional networks, there is a
great interest in generalizing convolutions to non-Euclidean manifolds. A major
complication in comparison to flat spaces is that it is unclear in which
alignment a convolution kernel should be applied on a manifold. The underlying
reason for this ambiguity is that general manifolds do not come with a
canonical choice of reference frames (gauge). Kernels and features therefore
have to be expressed relative to arbitrary coordinates. We argue that the
particular choice of coordinatization should not affect a network's inference
-- it should be coordinate independent. A simultaneous demand for coordinate
independence and weight sharing is shown to result in a requirement on the
network to be equivariant under local gauge transformations (changes of local
reference frames). The ambiguity of reference frames depends thereby on the
G-structure of the manifold, such that the necessary level of gauge
equivariance is prescribed by the corresponding structure group G. Coordinate
independent convolutions are proven to be equivariant w.r.t. those isometries
that are symmetries of the G-structure. The resulting theory is formulated in a
coordinate free fashion in terms of fiber bundles. To exemplify the design of
coordinate independent convolutions, we implement a convolutional network on
the M\"obius strip. The generality of our differential geometric formulation of
convolutional networks is demonstrated by an extensive literature review which
explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general
surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions
is publicly available at https://github.com/mauriceweiler/MobiusCNN