TBD. Various notes on a.e. continuous monotonic changes of index to align two process, and estimation thereof.

Warping, registration problems. Especially interesting for functional data analysis,

TODO: understand computational complexity of this. High, intuition suggests.

Mass transport problems - do these relate?

## Warping of point processes

A special interest of mine, at the intersection of functional data analysis, point processes and warping.

Though the study of multiple realisations of point processes has been considered prior to the emergence of FDA (see, e.g., Karr [22]), treating realisations of point processes as individual data objects within a functional data analysis context is a more recent development offering important advan- tages; a key paper is that of WuMZ13 (also see Chiou and Müller [10] and ChWH05). Such data may be an object of interest in themselves (see, e.g., WuMZ13, ArMü14, WuSr12) but may also arise as landmark data in an otherwise classical functional data analysis (see, e.g., GaKn95, ArMü14). The recent surge of interest is exemplified in an upcoming discussion paper by WuSr14, whose discussion documents early progress and challenges in the field.

## Other warps

Does the CTC warp-robust loss function (GFGS06) fit here?

CTC:

Connectionist Temporal Classification is a loss function useful for performing supervised learning on sequence data, without needing an alignment between input data and labels. For example, CTC can be used to train end-to-end systems for speech recognition, which is how we have been using it at Baidu’s Silicon Valley AI Lab.

## Refs

- AABC15
- Amodei, D., Anubhai, R., Battenberg, E., Case, C., Casper, J., Catanzaro, B., … Zhu, Z. (2015) Deep Speech 2: End-to-End Speech Recognition in English and Mandarin.
*arXiv:1512.02595 [cs]*. - ArMü14
- Arribas-Gil, A., & Müller, H.-G. (2014) Pairwise dynamic time warping for event data.
*Computational Statistics & Data Analysis*, 69, 255–268. DOI. - ArRo12
- Arribas-Gil, A., & Romo, J. (2012) Robust depth-based estimation in the time warping model.
*Biostatistics (Oxford, England)*, 13(3), 398–414. DOI. - BBVK02
- Brown, E., Barbieri, R., Ventura, V., Kass, R., & Frank, L. (2002) The time-rescaling theorem and its application to neural spike train data analysis.
*Neural Computation*, 14(2), 325–346. DOI. - CMTB14
- Caramiaux, B., Montecchio, N., Tanaka, A., & Bevilacqua, F. (2014) Adaptive Gesture Recognition with Variation Estimation for Interactive Systems.
*ACM Trans. Interact. Intell. Syst.*, 4(4), 18:1–18:34. DOI. - ChWH05
- Chiang, C.-T., Wang, M.-C., & Huang, C.-Y. (2005) Kernel Estimation of Rate Function for Recurrent Event Data.
*Scandinavian Journal of Statistics*, 32(1), 77–91. DOI. - FiHM00
- Fitzpatrick, J. M., Hill, D. L. G., & Maurer, Jr., C. R.(2000) Image Registration. In M. Sonka & J. M. Fitzpatrick (Eds.), Handbook of Medical Imaging, Volume 2. Medical Image Processing and Analysis (p. Chapter 8). 1000 20th Street, Bellingham, WA 98227-0010 USA: SPIE
- GaKn95
- Gasser, T., & Kneip, A. (1995) Searching for Structure in Curve Samples.
*Journal of the American Statistical Association*, 90(432), 1179–1188. DOI. - GiKO11
- Gillian, N., Knapp, B., & O’Modhrain, S. (2011) Recognition of multivariate temporal musical gestures using n-dimensional dynamic time warping.
- GKTN08
- Glocker, B., Komodakis, N., Tziritas, G., Navab, N., & Paragios, N. (2008) Dense image registration through MRFs and efficient linear programmingq.
*Medical Image Analysis*, 12(6), 731–741. DOI. - GSKP11
- Glocker, B., Sotiras, A., Komodakis, N., & Paragios, N. (2011) Deformable Medical Image Registration: Setting the State of the Art with Discrete Methods.
*Annual Review of Biomedical Engineering*, 13(1), 219–244. DOI. - GFGS06
- Graves, A., Fernández, S., Gomez, F., & Schmidhuber, J. (2006) Connectionist Temporal Classification: Labelling Unsegmented Sequence Data with Recurrent Neural Networks. In Proceedings of the 23rd International Conference on Machine Learning (pp. 369–376). New York, NY, USA: ACM DOI.
- Jame07
- James, G. M.(2007) Curve alignment by moments.
*The Annals of Applied Statistics*, 1(2), 480–501. DOI. - PaZe16
- Panaretos, V. M., & Zemel, Y. (2016) Separation of Amplitude and Phase Variation in Point Processes.
*The Annals of Statistics*, 44(2), 771–812. DOI. - WuMZ13
- Wu, S., Müller, H.-G., & Zhang, Z. (2013) Functional Data Analysis for Point Processes with Rare Events.
*Statistica Sinica*, 23(1), 1–23. - WuSr12
- Wu, W., & Srivastava, A. (2012) Estimating summary statistics in the spike-train space.
*Journal of Computational Neuroscience*, 34(3), 391–410. DOI. - WuSr14
- Wu, W., & Srivastava, A. (2014) Analysis of spike train data: Alignment and comparisons using the extended Fisher-Rao metric.
*Electronic Journal of Statistics*, 8(2), 1776–1785. DOI. - YiJF98
- Yi, B.-K., Jagadish, H. V., & Faloutsos, C. (1998) Efficient retrieval of similar time sequences under time warping. In , 14th International Conference on Data Engineering, 1998. Proceedings (pp. 201–208). DOI.