19 research outputs found
Beyond topological persistence: Starting from networks
Persistent homology enables fast and computable comparison of topological
objects. However, it is naturally limited to the analysis of topological
spaces. We extend the theory of persistence, by guaranteeing robustness and
computability to significant data types as simple graphs and quivers. We focus
on categorical persistence functions that allow us to study in full generality
strong kinds of connectedness such as clique communities, -vertex and
-edge connectedness directly on simple graphs and monic coherent categories.Comment: arXiv admin note: text overlap with arXiv:1707.0967
Parametric machines: a fresh approach to architecture search
Using tools from category theory, we provide a framework where artificial
neural networks, and their architectures, can be formally described. We first
define the notion of machine in a general categorical context, and show how
simple machines can be combined into more complex ones. We explore finite- and
infinite-depth machines, which generalize neural networks and neural ordinary
differential equations. Borrowing ideas from functional analysis and kernel
methods, we build complete, normed, infinite-dimensional spaces of machines,
and discuss how to find optimal architectures and parameters -- within those
spaces -- to solve a given computational problem. In our numerical experiments,
these kernel-inspired networks can outperform classical neural networks when
the training dataset is small.Comment: 31 pages, 4 figure
Steady and ranging sets in graph persistence
Generalised persistence functions (gp-functions) are defined on -indexed diagrams in a given category. A sufficient condition for
stability is also introduced. In the category of graphs, a standard way of
producing gp-functions is proposed: steady and ranging sets for a given
feature. The example of steady and ranging hubs is studied in depth; their
meaning is investigated in three concrete networks
Persistence-based operators in machine learning
Artificial neural networks can learn complex, salient data features to
achieve a given task. On the opposite end of the spectrum, mathematically
grounded methods such as topological data analysis allow users to design
analysis pipelines fully aware of data constraints and symmetries. We introduce
a class of persistence-based neural network layers. Persistence-based layers
allow the users to easily inject knowledge about symmetries (equivariance)
respected by the data, are equipped with learnable weights, and can be composed
with state-of-the-art neural architectures
Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning
The aim of this paper is to provide a general mathematical framework for
group equivariance in the machine learning context. The framework builds on a
synergy between persistent homology and the theory of group actions. We define
group-equivariant non-expansive operators (GENEOs), which are maps between
function spaces associated with groups of transformations. We study the
topological and metric properties of the space of GENEOs to evaluate their
approximating power and set the basis for general strategies to initialise and
compose operators. We begin by defining suitable pseudo-metrics for the
function spaces, the equivariance groups, and the set of non-expansive
operators. Basing on these pseudo-metrics, we prove that the space of GENEOs is
compact and convex, under the assumption that the function spaces are compact
and convex. These results provide fundamental guarantees in a machine learning
perspective. We show examples on the MNIST and fashion-MNIST datasets. By
considering isometry-equivariant non-expansive operators, we describe a simple
strategy to select and sample operators, and show how the selected and sampled
operators can be used to perform both classical metric learning and an
effective initialisation of the kernels of a convolutional neural network.Comment: Added references. Extended Section 7. Added 3 figures. Corrected
typos. 42 pages, 7 figure