# A Trainable Approach to Zero-delay Smoothing Spline Interpolation

arXiv: https://arxiv.org/abs/2203.03776 · HTML: https://mlsanity.com/paper/2203.03776/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 66 references that are in the map.
- First posted on arXiv 7 Mar 2022.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Recent Advances in Recurrent Neural Networks](https://mlsanity.com/paper/1801.01078/) (2017)
- [Incremental Ensemble Gaussian Processes](https://mlsanity.com/paper/2110.06777/) (2021)
- [Real-Time Machine Learning: The Missing Pieces](https://mlsanity.com/paper/1703.03924/) (2017)

## Built on by (most cited first, 1 of 1)

- [Consistent Signal Reconstruction from Streaming Multivariate Time Series](https://mlsanity.com/paper/2308.12459/) (2023)

## Abstract (arXiv)

The task of reconstructing smooth signals from streamed data in the form of signal samples arises in various applications. This work addresses such a task subject to a zero-delay response; that is, the smooth signal must be reconstructed sequentially as soon as a data sample is available and without having access to subsequent data. State-of-the-art approaches solve this problem by interpolating consecutive data samples using splines. Here, each interpolation step yields a piece that ensures a smooth signal reconstruction while minimizing a cost metric, typically a weighted sum between the squared residual and a derivative-based measure of smoothness. As a result, a zero-delay interpolation is achieved in exchange for an almost certainly higher cumulative cost as compared to interpolating all data samples together. This paper presents a novel approach to further reduce this cumulative cost on average. First, we formulate a zero-delay smoothing spline interpolation problem from a sequential decision-making perspective, allowing us to model the future impact of each interpolated piece on the average cumulative cost. Then, an interpolation method is proposed to exploit the temporal dependencies between the streamed data samples. Our method is assisted by a recurrent neural network and accordingly trained to reduce the accumulated cost on average over a set of example data samples collected from the same signal source generating the signal to be reconstructed. Finally, we present extensive experimental results for synthetic and real data showing how our approach outperforms the abovementioned state-of-the-art.