Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: This paper studies change-point detection and localization for sparsely or densely sampled functional data observed on a general d-dimensional space. A computational efficient kernel based FSBS algorithm is proposed to handle discretely observed functional data and general d-dimensional domain. The supporting theory is developed for heavy-tailed and serially dependent curves and measurement errors.  A phase transition phenomenon is established depending on the order of n relative to T. The finite sample performance is evaluated via both simulations under different settings and one real data example. 
strengths_and_weaknesses: Strength:

1, The first work in existing literature studying change points detection for partially observed functional data. 

2, Can handle low-dimensional multivariate functional data.

3, Theory is developed for serially dependent random functions and measurement errors

Weakness:

1, Instead of a machine learning conference like NeurIPS, I feel this paper fits a traditional Statistics journal better. The proposed methodology seems like a combination of some well-established techniques in change points detection and functional data analysis areas.

2, The measurement locations x_{t,i} are fixed, while in standard functional and longitudinal data analysis literature, these locations are assumed to be random and the number of measurement locations across curves can be different (rather than a fixed n) or even random.

3, The detailed comparison with phase transition phenomenon in existing literature is needed. See my question below.

4, Local constant smoother instead of local linear smoother is used and hence may suffer from bad performance around the boundaries. 

questions: 1, What is the advantage of the local constant smoother used in the paper instead of the commonly adopted local linear smoother?

2, The phase transition result is interesting, but is different from results in Zhang and Wang, 2016, which considers categories of sparse, dense and ultra-dense functional data. In particular, the optimal bandwidths are different under different cases and as the number of measurement locations becomes sufficiently large, the parametric root-n rate can be attained. A detailed discussion is helpful.

3, What are the truly innovative component of the paper taking the advantages of functional, temporally dependent and non-stationary nature of the data?

4, Why is the extreme sparse case n=1 argued as one selling point of the paper? As far as I know, if n is bounded (treated as sparse functional data), the same rate can be attained.
limitations: Yes
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper proposes a new kernal-based algorithm called functional seeded binary segmentation (FSBS) for change point detection of functional data. Theoretical error rate and consistency results are presented for functional data with general sampling density (sparse & dense), temporal dependence, and importantly that are multi-dimensional. Numerial experiments are conducted on simulation data as well as climate change data.
strengths_and_weaknesses: This paper is very clearly written and the presentation is great. While I haven't had a chance to read through the proofs in appendix, assuming they are technically sound, extending changepoint detection to multi-dimensional functional data is a significant contribution to the community.
questions: 1. Since the paper stresses extension to multi-dimensional data as a major contribution, there should be more discussion (in section 3.2) and experiments (in both section 4.1 and 4.2) involving multi-dimensional data. What insight can one gain about the theoretical bounds with respect to parameter $d$? How does the algorithm behave for $d >1$ in more than just a single experiment (scenario 4)?

2. How did the computational cost and runtime of the proposed method compare to the existing methods?

3. It's not clear what the reader is supposed to notice from Figure 2. If there is noticable change in the top left corner, a zoomed-in panel might be helpful.

limitations: N/A
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper considers the change-point detection problem for functional data in multi-dimensional space. A kernel-based algorithm is proposed, which allows for handling functional data in multivariate domain with measurement error. The author further presents several important theoretical results about this algorithm, such as the consistency of testing, and sharp localization error rate. Numerical results support their contributions. 
strengths_and_weaknesses: **Originality/Significance**: The proposed kernel-based algorithm elegantly solves main challenges in existing literature: (i) observed funtional data may contain measurement error, (ii) there may exist multiple change points, (iii) domain for functional data can be multivariate.  This work would be very interesting for readers in the testing and detection area.

**Quality**: All important points for functional-data change point detection are discussed, including the algorithm development, solid theoretical analysis, and extensive experiment simulations. From my view point, the quality of this paper is at least above 50% of all accepted nips papers.

**Clarity**: Problem setup, algorithm procedures, theory statements and assumptions are clearly presented. 

**Weakness**: 
1. It would be helpful to do a literature review for kernel-based change-point detection (even not under the functional data setting). This comparison will make the reader understand the connection and difference for the proposed algorithm.
2. The choice of kernel will largely influence the performance of change-point detection. The kernel choice in general is specified by the users that have prior knowledge on the data. Although the author does not demonstrate which kernel choice is better (in numerical simulation only Gaussian kernel is used), I think it is not a big issue and it would be an interesting topic for future study.
3. Minor wording issue: 
- In line 168, the bracket should be $O(T\log(T))$.
- In line 211-212, there is a large box for the equation environment. The author may consider repharse it to be "with probability at least $1-3\log^{-1}(T)$, it holds that \[xxx\]".
questions: See **Weakness** part.
limitations: See **Weakness** part.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper studies offline change-point detection and estimation for functional data. A kernel-based algorithm named functional seeded binary segmentation (FSBS) is proposed. This paper also shows the consistency of FSBS for multiple change-point estimation and 	.	provides a sharp localization error rate. Numerical demonstrations on synthetic and real data are provided.
strengths_and_weaknesses: Strengths

1. A new multiple change point detection algorithm is proposed. The FSBS procedure, based on kernel density estimation and CUSUM statistics, is computationally efficient. 
2. This paper considers a general detection scheme, where the functional data can be discretely observed (even with n=1 in the very extreme case) and can be contaminated with temporally-dependent and heavy-tailed errors.
3. Theoretical guarantee on the estimation error is provided. 
4. Extensive numerical experiments 


Weaknesses
The method is not new. Both Binary Segmentation and kernel density estimations are well-studied methods for change detection and nonparametric density estimation, respectively. And the theoretical guarantee on BS and KDE are also well-studied.

questions: Some notations are not explained sufficiently. For example, what is the meaning of “arbitrarily-slow” in assumption 3, and what does C_SNR mean here?  

Also, considering kernel-based change-point detection, it might be worthwhile to compare to or comment on the difference with some existing kernel type methods, such as "Li, S., Xie, Y., Dai, H., & Song, L. (2015). M-statistic for kernel change-point detection. Advances in Neural Information Processing Systems, 28." (although this paper didn't target detection for functional data, its online detection scheme might be helpful if the FSBS could be generalized for online detection).

Minor issues:
The equation in Theorem 1 is too long, maybe split into two lines.

limitations: yes
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper presented a change-point detection method, namely functional seeded binary segmentation (FSBS), for d-dimensional time-series data. FSBS is claimed to be more suitable for heavy-tailed and temporally dependent data, while being computationally efficient. The proposed method is evaluated on both synthetic data and sea surface temperature data and leads to better results compared to three classic methods: BGHK, HK, and SN.
strengths_and_weaknesses: * Strengths

-- The proposed method is easy to implement with fewer (only three) tuning parameters, and the authors explained them well in terms of how to the select them.

-- The authors showed that FSBS is consistent in detecting and localising multiple change-points under standard regularity conditions. Besides, it is claimed to have the sharpest localisation error rate.

* Weaknesses

-- The baseline methods appear to be published quit a while ago, e.g., BGHK (2009), HK (2010) and SN (2011), so I’m not sure if they are state-of-the-art methods for change-point detection in functional data as the authors claimed in Line 267.

-- The experiments seem to be quite lightweight, as it has only one real dataset.

questions: FSBS is claimed to handle temporally-dependent observations better theoretically, however, it is not clearly why it has such a property and other methods do not. In line 128 – 130, authors state that “For temporal dependence, we adopt the physical dependence framework by [26], which covers a wide range of time series models, such as ARMA and vector AR models.” while it is hard to tell how [26] is adopted here exactly.
limitations: See “weakness” above.
review_title: null

Meta-review: metareview: The paper studies change-point detection and localization for functional data, which is an interesting and timely topic. I agree with some reviewers that the paper might be a better fit with the traditional statistical venue. The authors have done a great job in the rebuttal phase in addressing reviewers’ comments. I believe it is a worthwhile paper to be published in NeurIPS.
confidence: Certain
Decision: Accept