110 research outputs found
Polyhedral computational geometry for averaging metric phylogenetic trees
This paper investigates the computational geometry relevant to calculations
of the Frechet mean and variance for probability distributions on the
phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature developed by Sturm. We
show that the combinatorics of geodesics with a specified fixed endpoint in
tree space are determined by the location of the varying endpoint in a certain
polyhedral subdivision of tree space. The variance function associated to a
finite subset of tree space has a fixed algebraic formula within
each cell of the corresponding subdivision, and is continuously differentiable
in the interior of each orthant of tree space. We use this subdivision to
establish two iterative methods for producing sequences that converge to the
Frechet mean: one based on Sturm's Law of Large Numbers, and another based on
descent algorithms for finding optima of smooth functions on convex polyhedra.
We present properties and biological applications of Frechet means and extend
our main results to more general globally nonpositively curved spaces composed
of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5,
added counter example for polyhedrality of vistal subdivision in general
CAT(0) cubical complexes; v1: 43 pages, 5 figure
Faster Algorithms for Largest Empty Rectangles and Boxes
We revisit a classical problem in computational geometry: finding the
largest-volume axis-aligned empty box (inside a given bounding box) amidst
given points in dimensions. Previously, the best algorithms known have
running time for (by Aggarwal and Suri [SoCG'87]) and near
for . We describe faster algorithms with running time (i)
for , (ii) time for ,
and (iii) time for any constant .
To obtain the higher-dimensional result, we adapt and extend previous
techniques for Klee's measure problem to optimize certain objective functions
over the complement of a union of orthants.Comment: full version of a SoCG 2021 pape
4D Dual-Tree Complex Wavelets for Time-Dependent Data
The dual-tree complex wavelet transform (DT-ℂWT) is extended to the 4D setting. Key properties of 4D DT-ℂWT, such as directional sensitivity and shift-invariance, are discussed and illustrated in a tomographic application. The inverse problem of reconstructing a dynamic three-dimensional target from X-ray projection measurements can be formulated as 4D space-time tomography. The results suggest that 4D DT-ℂWT offers simple implementations combined with useful theoretical properties for tomographic reconstruction.Peer reviewe
What Can Transformers Learn In-Context? A Case Study of Simple Function Classes
In-context learning refers to the ability of a model to condition on a prompt
sequence consisting of in-context examples (input-output pairs corresponding to
some task) along with a new query input, and generate the corresponding output.
Crucially, in-context learning happens only at inference time without any
parameter updates to the model. While large language models such as GPT-3
exhibit some ability to perform in-context learning, it is unclear what the
relationship is between tasks on which this succeeds and what is present in the
training data. To make progress towards understanding in-context learning, we
consider the well-defined problem of training a model to in-context learn a
function class (e.g., linear functions): that is, given data derived from some
functions in the class, can we train a model to in-context learn "most"
functions from this class? We show empirically that standard Transformers can
be trained from scratch to perform in-context learning of linear functions --
that is, the trained model is able to learn unseen linear functions from
in-context examples with performance comparable to the optimal least squares
estimator. In fact, in-context learning is possible even under two forms of
distribution shift: (i) between the training data of the model and
inference-time prompts, and (ii) between the in-context examples and the query
input during inference. We also show that we can train Transformers to
in-context learn more complex function classes -- namely sparse linear
functions, two-layer neural networks, and decision trees -- with performance
that matches or exceeds task-specific learning algorithms. Our code and models
are available at https://github.com/dtsip/in-context-learning
- …