12 research outputs found
Invariants of multidimensional time series based on their iterated-integral signature
We introduce a novel class of features for multidimensional time series, that
are invariant with respect to transformations of the ambient space. The general
linear group, the group of rotations and the group of permutations of the axes
are considered. The starting point for their construction is Chen's
iterated-integral signature.Comment: complete rewrite of Section 3.
Iterated-integral signatures in machine learning
The iterated-integral signature, or rough-path signature, of a path has proved useful in several machine learning applications in the last few years. This work is extended in a number of ways. Algorithms for computing the signature and log signature efficiently are investigated and evaluated, which is useful for many applications of signatures when working with large datasets. Online Chinese character recognition using signature features with recurrent neural networks is investigated. A recurrent neural network cell which stores its memory as the signature of a path is suggested and demonstrated on a toy problem.
There is an essentially unique element of the signature of a path in space which, under transformations of the space, scales with volume. That element is characterised geometrically.
Given two features of curves, you can make a new one by taking the signed area of the 2d curve those two features make as a curve is traced out. A simple algebraic description of those features (which turn out to be signature elements) which can be formed from linear combinations of such combinations of total displacements is conjectured and worked towards. This is know as “areas of areas”
A structure theorem for streamed information
We identify the free half shuffle algebra of Sch\"utzenberger (1958) with an
algebra of real-valued functionals on paths, where the half shuffle emulates
integration of a functional against another. We then provide two, to our
knowledge, new identities in arity 3 involving its commutator (area), and show
that these are sufficient to recover the Zinbiel and Tortkara identities of
Dzhumadil'daev (2007). We use these identities to prove that any element of the
free half shuffle algebra can be expressed as a polynomial over iterated areas.
Moreover, we consider minimal sets of iterated integrals defined through the
recursive application of the half shuffle on Hall trees. Leveraging the duality
between this set of Hall integrals and classical Hall bases of the free Lie
algebra, we prove using combinatorial arguments that any element of the free
half shuffle algebra can be written uniquely as a polynomial over Hall
integrals. We interpret this result as a structure theorem for streamed
information, loosely analogous to the unique prime factorisation of integers,
allowing to split any real valued function on streamed data into two parts: a
first that extracts and packages the streamed information into recursively
defined atomic objects (Hall integrals), and a second that evaluates a
polynomial function in these objects without further reference to the original
stream. The question of whether a similar result holds if Hall integrals are
replaced by Hall areas is left as an open conjecture. Finally, we construct a
canonical, but to our knowledge, new decomposition of the free half shuffle
algebra as shuffle power series in the greatest letter of the original alphabet
with coefficients in a sub-algebra freely generated by a new alphabet with an
infinite number of letters. We use this construction to provide a second proof
of our structure theorem
A structure theorem for streamed information
We identify the free half shuffle algebra of Schützenberger [31] with an algebra of real-valued functionals on paths, where the half shuffle emulates the integration of a functional against another. We then provide two, to our knowledge, new identities in arity 3 involving its commutator (area), and show that these are sufficient to recover the Zinbiel and Tortkara identities introduced by Dzhumadil'daev [11]. We then use these identities to provide a simple proof of the main result of Diehl et al. [8], namely that any element of the free half shuffle algebra can be expressed as a polynomial over iterated areas.
Moreover, we consider minimal sets of Hall iterated integrals defined through the recursive application of the half shuffle product to Hall trees. Leveraging the duality between this set of Hall integrals and classical Hall bases of the free Lie algebra, we prove using combinatorial arguments that any element of the free half shuffle algebra can be written uniquely as a polynomial over Hall integrals. We interpret this result as a structure theorem for streamed information, loosely analogous to the unique prime factorisation of integers, allowing to split any real valued function on streamed data into two parts: a first that extracts and packages the streamed information into recursively defined atomic objects (Hall integrals), and a second that evaluates a polynomial function in these objects without further reference to the original stream. The question of whether a similar result holds if Hall integrals are replaced by Hall areas is left as an open conjecture.
Finally, we construct a canonical, but to our knowledge, new decomposition of the free half shuffle algebra as shuffle power series in the greatest letter of the original alphabet with coefficients in a sub-algebra freely generated by a new alphabet with an infinite number of letters. We use this construction to provide a second proof of our structure theorem
Areas of areas generate the shuffle algebra
We consider the anti-symmetrization of the half-shuffle on words, which we
call the 'area' operator, since it corresponds to taking the signed area of
elements of the iterated-integral signature. The tensor algebra is a so-called
Tortkara algebra under this operator. We show that the iterated application of
the area operator is sufficient to recover the iterated-integral signature of a
path. Just as the "information" the second level adds to the first one is known
to be equivalent to the area between components of the path, this means that
all the information added by subsequent levels is equivalent to iterated areas.
On the way to this main result, we characterize (homogeneous) generating sets
of the shuffle algebra. We finally discuss compatibility between the area
operator and discrete integration and stochastic integration and conclude with
some results on the linear span of the areas of areas.Comment: added examples and remarks, corrected semimartingale/martingale par
Common Pets in 3D: Dynamic New-View Synthesis of Real-Life Deformable Categories
Obtaining photorealistic reconstructions of objects from sparse views is
inherently ambiguous and can only be achieved by learning suitable
reconstruction priors. Earlier works on sparse rigid object reconstruction
successfully learned such priors from large datasets such as CO3D. In this
paper, we extend this approach to dynamic objects. We use cats and dogs as a
representative example and introduce Common Pets in 3D (CoP3D), a collection of
crowd-sourced videos showing around 4,200 distinct pets. CoP3D is one of the
first large-scale datasets for benchmarking non-rigid 3D reconstruction "in the
wild". We also propose Tracker-NeRF, a method for learning 4D reconstruction
from our dataset. At test time, given a small number of video frames of an
unseen object, Tracker-NeRF predicts the trajectories of its 3D points and
generates new views, interpolating viewpoint and time. Results on CoP3D reveal
significantly better non-rigid new-view synthesis performance than existing
baselines
African Drought Risk Pay-Out Benchmarking
This report contains exploratory data analysis of rainfall and Water Resource Sufficiency Index (WRSI) data provided by African Risk Ca- pacity (ARC). The purpose is to assess the predictability of droughts in Africa. We assess the appropriateness of the historical WRSI bench- marks set by ARC members compared to the observed WRSI values for different regions. We conclude that the benchmarks are broadly sensible. We then compare a number of linear time series models based on their ability to fit and forecast the WRSI time series. We conclude that sim- pler models like Simple Moving Average and Moving Median are more appropriate than more sophisticated models containing trends and sea- sonality like Holt Winter and TBATS. We also investigate the use of the SARIMA and TBATS models to forecast the seasonal patterns observed in rainfall data and conclude that both models can generate structured forecasts that reflect seasonal variability. The statistical evidence how- ever favoured TBATS over SARIMA. Attempts to measure the influence of the El Nin ̃o-Southern Oscillation on rainfall levels are inconclusive for the areas studied. Finally we perform a simple application of univariate Extreme Value Theory to rainfall data and conclude that further inves- tigation is necessary to understand how the catastrophic famine that affected Ethiopia in the early 1980’s would be reflected in the data if a similar event were to reoccur today