750 research outputs found
Learning Laplacian Matrix in Smooth Graph Signal Representations
The construction of a meaningful graph plays a crucial role in the success of
many graph-based representations and algorithms for handling structured data,
especially in the emerging field of graph signal processing. However, a
meaningful graph is not always readily available from the data, nor easy to
define depending on the application domain. In particular, it is often
desirable in graph signal processing applications that a graph is chosen such
that the data admit certain regularity or smoothness on the graph. In this
paper, we address the problem of learning graph Laplacians, which is equivalent
to learning graph topologies, such that the input data form graph signals with
smooth variations on the resulting topology. To this end, we adopt a factor
analysis model for the graph signals and impose a Gaussian probabilistic prior
on the latent variables that control these signals. We show that the Gaussian
prior leads to an efficient representation that favors the smoothness property
of the graph signals. We then propose an algorithm for learning graphs that
enforces such property and is based on minimizing the variations of the signals
on the learned graph. Experiments on both synthetic and real world data
demonstrate that the proposed graph learning framework can efficiently infer
meaningful graph topologies from signal observations under the smoothness
prior
Structural Analysis of Network Traffic Matrix via Relaxed Principal Component Pursuit
The network traffic matrix is widely used in network operation and
management. It is therefore of crucial importance to analyze the components and
the structure of the network traffic matrix, for which several mathematical
approaches such as Principal Component Analysis (PCA) were proposed. In this
paper, we first argue that PCA performs poorly for analyzing traffic matrix
that is polluted by large volume anomalies, and then propose a new
decomposition model for the network traffic matrix. According to this model, we
carry out the structural analysis by decomposing the network traffic matrix
into three sub-matrices, namely, the deterministic traffic, the anomaly traffic
and the noise traffic matrix, which is similar to the Robust Principal
Component Analysis (RPCA) problem previously studied in [13]. Based on the
Relaxed Principal Component Pursuit (Relaxed PCP) method and the Accelerated
Proximal Gradient (APG) algorithm, we present an iterative approach for
decomposing a traffic matrix, and demonstrate its efficiency and flexibility by
experimental results. Finally, we further discuss several features of the
deterministic and noise traffic. Our study develops a novel method for the
problem of structural analysis of the traffic matrix, which is robust against
pollution of large volume anomalies.Comment: Accepted to Elsevier Computer Network
Understanding stock market instability via graph auto-encoders
Understanding stock market instability is a key question in financial
management as practitioners seek to forecast breakdowns in asset co-movements
which expose portfolios to rapid and devastating collapses in value. The
structure of these co-movements can be described as a graph where companies are
represented by nodes and edges capture correlations between their price
movements. Learning a timely indicator of co-movement breakdowns (manifested as
modifications in the graph structure) is central in understanding both
financial stability and volatility forecasting. We propose to use the edge
reconstruction accuracy of a graph auto-encoder (GAE) as an indicator for how
spatially homogeneous connections between assets are, which, based on financial
network literature, we use as a proxy to infer market volatility. Our
experiments on the S&P 500 over the 2015-2022 period show that higher GAE
reconstruction error values are correlated with higher volatility. We also show
that out-of-sample autoregressive modeling of volatility is improved by the
addition of the proposed measure. Our paper contributes to the literature of
machine learning in finance particularly in the context of understanding stock
market instability.Comment: Submitted to Glinda workshop of the Neurips 2022 conference Keywords
: Graph Based Learning, Graph Neural Networks, Graph Autoencoder, Stock
Market Information, Volatility Forecastin
Learning Hypergraphs From Signals With Dual Smoothness Prior
The construction of a meaningful hypergraph topology is the key to processing
signals with high-order relationships that involve more than two entities.
Learning the hypergraph structure from the observed signals to capture the
intrinsic relationships among the entities becomes crucial when a hypergraph
topology is not readily available in the datasets. There are two challenges
that lie at the heart of this problem: 1) how to handle the huge search space
of potential hyperedges, and 2) how to define meaningful criteria to measure
the relationship between the signals observed on nodes and the hypergraph
structure. In this paper, to address the first challenge, we adopt the
assumption that the ideal hypergraph structure can be derived from a learnable
graph structure that captures the pairwise relations within signals. Further,
we propose a hypergraph learning framework with a novel dual smoothness prior
that reveals a mapping between the observed node signals and the hypergraph
structure, whereby each hyperedge corresponds to a subgraph with both node
signal smoothness and edge signal smoothness in the learnable graph structure.
Finally, we conduct extensive experiments to evaluate the proposed framework on
both synthetic and real world datasets. Experiments show that our proposed
framework can efficiently infer meaningful hypergraph topologies from observed
signals.Comment: We have polished the paper and fixed some typos and the correct
number of the target hyperedges is given to the baseline in this versio
Hypergraph Structure Inference From Data Under Smoothness Prior
Hypergraphs are important for processing data with higher-order relationships
involving more than two entities. In scenarios where explicit hypergraphs are
not readily available, it is desirable to infer a meaningful hypergraph
structure from the node features to capture the intrinsic relations within the
data. However, existing methods either adopt simple pre-defined rules that fail
to precisely capture the distribution of the potential hypergraph structure, or
learn a mapping between hypergraph structures and node features but require a
large amount of labelled data, i.e., pre-existing hypergraph structures, for
training. Both restrict their applications in practical scenarios. To fill this
gap, we propose a novel smoothness prior that enables us to design a method to
infer the probability for each potential hyperedge without labelled data as
supervision. The proposed prior indicates features of nodes in a hyperedge are
highly correlated by the features of the hyperedge containing them. We use this
prior to derive the relation between the hypergraph structure and the node
features via probabilistic modelling. This allows us to develop an unsupervised
inference method to estimate the probability for each potential hyperedge via
solving an optimisation problem that has an analytical solution. Experiments on
both synthetic and real-world data demonstrate that our method can learn
meaningful hypergraph structures from data more efficiently than existing
hypergraph structure inference methods
Laplacian-regularized graph bandits: Algorithms and theoretical analysis
We consider a stochastic linear bandit problem with multiple users, where the
relationship between users is captured by an underlying graph and user
preferences are represented as smooth signals on the graph. We introduce a
novel bandit algorithm where the smoothness prior is imposed via the
random-walk graph Laplacian, which leads to a single-user cumulative regret
scaling as with time horizon ,
feature dimensionality , and the scalar parameter that
depends on the graph connectivity. This is an improvement over
in \algo{LinUCB}~\Ccite{li2010contextual},
where user relationship is not taken into account. In terms of network regret
(sum of cumulative regret over users), the proposed algorithm leads to a
scaling as , which is a significant
improvement over in the state-of-the-art
algorithm \algo{Gob.Lin} \Ccite{cesa2013gang}. To improve scalability, we
further propose a simplified algorithm with a linear computational complexity
with respect to the number of users, while maintaining the same regret.
Finally, we present a finite-time analysis on the proposed algorithms, and
demonstrate their advantage in comparison with state-of-the-art graph-based
bandit algorithms on both synthetic and real-world data
- …