413 research outputs found
Zero Shot Learning with the Isoperimetric Loss
We introduce the isoperimetric loss as a regularization criterion for
learning the map from a visual representation to a semantic embedding, to be
used to transfer knowledge to unknown classes in a zero-shot learning setting.
We use a pre-trained deep neural network model as a visual representation of
image data, a Word2Vec embedding of class labels, and linear maps between the
visual and semantic embedding spaces. However, the spaces themselves are not
linear, and we postulate the sample embedding to be populated by noisy samples
near otherwise smooth manifolds. We exploit the graph structure defined by the
sample points to regularize the estimates of the manifolds by inferring the
graph connectivity using a generalization of the isoperimetric inequalities
from Riemannian geometry to graphs. Surprisingly, this regularization alone,
paired with the simplest baseline model, outperforms the state-of-the-art among
fully automated methods in zero-shot learning benchmarks such as AwA and CUB.
This improvement is achieved solely by learning the structure of the underlying
spaces by imposing regularity.Comment: Accepted to AAAI-2
On the Smallest Eigenvalue of Grounded Laplacian Matrices
We provide upper and lower bounds on the smallest eigenvalue of grounded
Laplacian matrices (which are matrices obtained by removing certain rows and
columns of the Laplacian matrix of a given graph). The gap between the upper
and lower bounds depends on the ratio of the smallest and largest components of
the eigenvector corresponding to the smallest eigenvalue of the grounded
Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently
obtain a tight characterization of the smallest eigenvalue for certain classes
of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a
(sufficiently small) set of rows and columns is removed from the Laplacian,
and the probability of adding an edge is sufficiently large, the smallest
eigenvalue of the grounded Laplacian asymptotically almost surely approaches
. We also show that for random -regular graphs with a single row and
column removed, the smallest eigenvalue is . Our bounds
have applications to the study of the convergence rate in continuous-time and
discrete-time consensus dynamics with stubborn or leader nodes
Iterative solution of spatial network models by subspace decomposition
We present and analyze a preconditioned conjugate gradient method (PCG) for
solving spatial network problems. Primarily, we consider diffusion and
structural mechanics simulations for fiber based materials, but the methodology
can be applied to a wide range of models, fulfilling a set of abstract
assumptions. The proposed method builds on a classical subspace decomposition
into a coarse subspace, realized as the restriction of a finite element space
to the nodes of the spatial network, and localized subspaces with support on
mesh stars. The main contribution of this work is the convergence analysis of
the proposed method. The analysis translates results from finite element
theory, including interpolation bounds, to the spatial network setting. A
convergence rate of the PCG algorithm, only depending on global bounds of the
operator and homogeneity, connectivity and locality constants of the network,
is established. The theoretical results are confirmed by several numerical
experiments.Comment: Journal article draft, not peer-reviewe
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