This paper addresses learning end-to-end models for time series data that
include a temporal alignment step via dynamic time warping (DTW). Existing
approaches to differentiable DTW either differentiate through a fixed warping
path or apply a differentiable relaxation to the min operator found in the
recursive steps used to solve the DTW problem. We instead propose a DTW layer
based around bi-level optimisation and deep declarative networks, which we name
DecDTW. By formulating DTW as a continuous, inequality constrained optimisation
problem, we can compute gradients for the solution of the optimal alignment
(with respect to the underlying time series) using implicit differentiation. An
interesting byproduct of this formulation is that DecDTW outputs the optimal
warping path between two time series as opposed to a soft approximation,
recoverable from Soft-DTW. We show that this property is particularly useful
for applications where downstream loss functions are defined on the optimal
alignment path itself. This naturally occurs, for instance, when learning to
improve the accuracy of predicted alignments against ground truth alignments.
We evaluate DecDTW on two such applications, namely the audio-to-score
alignment task in music information retrieval and the visual place recognition
task in robotics, demonstrating state-of-the-art results in both.

本文提出了一种基于双层优化和深度声明网络的 DTW 层，可以将 DTW 表示为连续的优化问题，并计算出最优的对齐路径，可用于自动化的时间序列数据对齐。

用于端到端对齐路径学习的深度声明动态时间规整

Deep Declarative Dynamic Time Warping for End-to-End Learning of  Alignment Paths

We explore a new class of end-to-end learnable models wherein data processing
nodes (or network layers) are defined in terms of desired behavior rather than
an explicit forward function. Specifically, the forward function is implicitly
defined as the solution to a mathematical optimization problem. Consistent with
nomenclature in the programming languages community, we name these models deep
declarative networks. Importantly, we show that the class of deep declarative
networks subsumes current deep learning models. Moreover, invoking the implicit
function theorem, we show how gradients can be back-propagated through many
declaratively defined data processing nodes thereby enabling end-to-end
learning. We show how these declarative processing nodes can be implemented in
the popular PyTorch deep learning software library allowing declarative and
imperative nodes to co-exist within the same network. We also provide numerous
insights and illustrative examples of declarative nodes and demonstrate their
application for image and point cloud classification tasks.

本研究探索了一类新的端到端可学习模型，其中数据处理节点是根据预期行为而不是显式的前向函数定义的，并且演示了这些模型如何超越当前的深度学习模型，如何通过隐式函数定理进行反向传播导致端到端学习，以及如何在常见的 PyTorch 深度学习软件库中实现这些声明性处理节点。