Exploiting Representation Curvature for Boundary Detection in Time Series
Authors: Yooju Shin
Jaehyun Park
Susik Yoon
Hwanjun Song
Byung Suk Lee
Jae-Gil Lee
Venue: NeurIPS 2024 poster (2024)
Keywords: time series
representation
boundary detection
Primary area: machine_learning_for_other_sciences_and_fields
TL;DR: We propose RECURVE that exploits representation curvature for time-series boundary detection.

Abstract: *Boundaries* are the timestamps at which a class in a time series changes. Recently, representation-based boundary detection has gained popularity, but its emphasis on consecutive distance difference backfires, especially when the changes are gradual. In this paper, we propose a boundary detection method, **RECURVE**, based on a novel change metric, the ***curvature*** of a representation trajectory, to accommodate both gradual and abrupt changes. Here, a sequence of representations in the representation space is interpreted as a trajectory, and a curvature at each timestamp can be computed. Using the theory of random walk, we formally show that the mean curvature is lower near boundaries than at other points. Extensive experiments using diverse real-world time-series datasets confirm the superiority of RECURVE over state-of-the-art methods.

Exploiting Representation Curvature for Boundary Detection in Time Series
Yooju Shin', Jaehyun Park', Hwanjun Song', Susik Yoon', Byung Suk Lee", JJa--Gil Lee
KAIST, Koree University, UUniversity of Vermont
(yooju. shin, jhpark813, jaegil, songhwanjun) @kaist.ac.krr susikōkorea, ac kr, bsleeduvm.edi
Abstract
Boundaries the timestamps which class in time series changes. Recently, representation-but based boundary detection has gained popularity, but its emphasis on consecutive distance difference backfires, especially when the changes are gradual. In this paper, we propose boundary detection method, RECURVE. based on novel change metric, the curvature of a representation trajectory, accommodate both gradual and abrupt changes. Here, a sequence of representations in the representation space is interpreted as trajectory, and curvature at each timestamp can be computed. Using the theory of random walk, we formally show that the mean curvature is lower near boundaries than at other points. Extensive experiments using diverse real-world time-series datasets confirm the superiority RECURVE over state-of-the-art methods.
Introduction
In a time series composed of sequential data points (simply points) indexed by timestamps, there are boundaries (or change points) signifying transitions between different classes or states, such as a shift from running to walking [1 2]. Detecting boundaries is a crucial task preprocessing and diverse applications of time-series data. As preprocessing, they partition time series into segments of coherent points, accelerating annotation of the time-series for further analysis and giving additional supervision in classification
6 77 As primary tasks, they are valuable for identifying changes that require human attention in a variety of domains, including climate, health care, finance, and manufacturing: epilepsy detection, stock price tracking, and action segmentation examples of
4
5
possible applications 8 10, II
Representation based boundary detection methods [12 13 are prevalent today because they do not require specific assumptions on time-series properties, such as distribution or temporal shape, and can handle high dimensionality due to the capability of self-supervised model that autonomously learns distinctive features from raw time series without any supervision.
these methods, model [14 15 16 is first used to derive representation of each point, and then point is identified as a boundary its representation significantly deviates from those of adjacent points. Let's refer the points close to boundary inter-segment points and the remaining points intra-segment points. In short, these methods operate by assuming that the distance between consecutive representations greater between inter-segment points than between intra-segment points.
supervised
However, this assumption on the distance difference does not always hold, especially when the changes are subtle or gradual. Time-series representation learning methods often pursue preserving the temporal coherence of a time series as their training goal to make temporally close points similar in their representations and distant points dissimilar [14]. As demonstrated in Figure I the consecutive distances are not clearly distinguishable between intra- and inter-segment points for relatively subtle changes with stair up t-> stair down because just the direction of motion differs
*Correspondngg author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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5
High
Up or down
Stand or sit
Density
Up e- down
Stand
Inter
Inira
Large
Small overlap
Intra
overlap
Low CUIV.
Class ball
0.6
0.8
1.0
0.6
0.8
1.0
Consecutive cosine similarity
Representation Space
Figure I: Consecutive distance (cosine similarity) distribu- tion from intra- and inter-segment points in the representa- tions of the HAPT dataset.
Curvature comparison be- tween intra- and inter-segment points in representation space.
Figure 2:
between the two classes, whereas they are for abrupt changes with stand + sit. Thus, the inability handle subtle changes hinders achieving an overall good performance.
In this work, we take a novel perspective on detecting boundaries by leveraging curvatures instead of distances in the representation space. As shown in Figure 2 the curvature point in curve measures the instantaneous rate of direction change, or more precisely, the amount by which the curve deviates from being straight line 17.. Suppose that a sequence of point representations from time series constitutes a representation trajectory. We observe that, regardless of whether the changes gradual abrupt, the direction of the representation trajectory tends change more sharply (showing a higher curvature) at intra-segment points than at inter segment points. Accordingly. we contend that the curvature of a representation trajectory should be very promising indicator for class boundary detection.
Using Figure 2] we justify the intuition behind curvature-based boundary detection. Because representation learning tries to learn class-separated features, well-embedded points a certain class (or segment) be drawn from its class-specific ball [18 [9]. That is, the representation trajectory of intra-segment points is confined within a class ball, whereas that of inter-segment points is not, Then, for intra-segment points to reside exclusively within class ball, their representation trajectory needs to make sharp turns frequently. In contrast, the transition from one class ball to another does not necessarily make sharp turns. This observation formally proven by the relationship between the mean curvature and the radius of a confining hypersphere, assuming a random walk of a point representation (see Section 3.4..
Overall, for boundary detection, treating sequence of representations as a trajectory and measuring its curvature is an entirely novel approach, which results in RECURVE (Representation trajEctory CURVaturE). A representation trajectory is derived by a time-series representation learning method, and the curvature at each point is calculated very efficiently; then, the points whose curvature is relatively small are identified as boundaries. RECURVE is simple yet powerful, and can be combined with any time-series representation learning method. We conduct comprehensive evaluations variety of time-series datasets, comparing it against statc-of-the-arr boundary detection methods. The results demonstrate that RECURVE consistently enhances the accuracy, achieving improvements of up to 12.7%. Furthermore, this superiority is shown to exist regardless of the degree of change between different classes.
2 Related Work
2.1 Boundary Detection
Time-series boundary detection methods assess the dissimilarity between two successive intervals and apply a threshold to pinpoint the positions of boundarics. There are multiple methods available for quantifying dissimilarity: (1) conducting statistical tests, (2) quantifying the deviation from discovered patterns, and (3) calculating distances between the representations learned from self- supervised model. We summarize each category here, with additional in-depth details available in extensive surveys
Statistical tests often rely on the probability density ratio of two consecutive intervals as a key statistic. CUSUM a traditional parametric algorithm that adds up the log likelihood ratio when a probability density function is given 20) 21 Rul.SIF is non-parametric algorithm that directly estimates the
probability density ratio using Pearson divergence without a probability density function [22 23
A kernel- based statistical test maps each interval to kernel space and then computes the kernel

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Fisher discriminant ratio as statistic [25]. KL-CPD uses deep neural network as a generator for kernel parameters, which solves high sensitivity in selecting parameters [26].
The proactive diseovery of frequent temporal patterns necessary for temporal pattern-hased bound- ary detection. FL.SS stores the locations of similar subsequences in time series using Matrix Profile and measures the likelihood of a regime change [27]. Motif-based boundary detection relies on the identification of short temporal patterns (motif's) determined through the minimum description length criterion; these motifss are then compared for similarity with other subsequences within a time series [T] 28 ESPRESSO, on the other hand, is hybrid of patter- and statistic- based approaches, detecting a wide range of boundaries across different scenarios and data types [29].
Representation-basee boundary detection methods are distinguished by the manner in which self-supervised model is trained. TIRE exploits an autoencoder to retain time-invariant features in consecutive timestamps to make representations of boundaries salient 12]; after training. the output representations undergo a process of smoothing, wherein a moving average is applied prior to the dissimilarity computation. TS-CP² leverages contrastive learning techniques to promote close proximity between representations of two consecutive timestamps and distant proximity between representations at randomly selected timestamps [3]; it examines the difference between each consecutive distance and the moving average.
2.2 Time-Series Representation Learning
Time-series representation learning builds a model create versatile representations capable of performing diverse downstream tasks such as classification, forecasting, and anomaly detection [0) 31. Reconstruction- based learning methods train autoencoder-baate deep neural networks using a reconstruction loss. TimeNet is an early example that uses a requence-to-sequeee
autoencoder
and uses the hidden embedding extracted from the encoder representation (2] DTTC extends traditional reconstruction-basee
learning by incorporating k-means loss alongside the reconstruction
loss (33) Input masking is also commonly used for reconstructing data with specific timestamps
intentionally masked or hidden 34 35 36
In contrastive learning, the Info-NCE (Noise Contrastive Estimation) loss plays pivotal role by bringing a positive pair closer together and pushing a negative pair apart in the representation space. An early approach considers sampled window and subsequence from the window positive pair (37. In recent methods such as TNC (Temporal Neighborhood Coding), the temporal distance serves a criterion for identifying positive pair, keeping two neighboring timestamp representations Following the principles of SimCIR RR [39], a positive pair can be created by pairing sampled window with its augmentation which involves data perturbation context changes [15 40). Besides, the Fourier transform of a time series serves as an augmentation technique for generating positive pairs or providing a new representation space []6 42
close 13 14 38
3 RECURVE: Curvature-based Boundary Detection
3.1 Preliminaries and Problem Setting
Dataset and Model: Let X' (xx)T=1 be time series. where T is the total number of points, and E Rdd is a d-dimensional point at timestamp t. Let C C (4x k E [1, K]] be set of the timestamps for the ground-truth boundaries. Considering class labels annotated each timestamp, C is composed of the timestamps where there is a change in the label from the previous one (e.g.. stand - walk). A window Xt,,. (mI)) is sequence of consecutive 2/ points centered at timestamp Im- A representation model fo. which deep neural network parameterized by converts each window X,,, to its representation Zt... E Rdd i.c., Zin.. = fo(......
Representation Learning: RECURVE is not bound to a specific representation learning method, and we summarize the training process using one of the popular methods, the temporal predictive coding (TPC) proposed in T [3] Here, two non-overlapping consecutive windows are used as a positive pair, and two randomly-sampled windows are used as negative pair. Thus, TS- Cp2 randomly samples b windows as well as their succeeding windows and constructs batch X ): the method subsequently minimizes the B = (X,,; .i.....: Xty, Xt +2,, X N1gt21.-...
InfoNCE loss (NCE [43]
3

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1 INE
Σ log
exp(sim(zz, 24, #2)+2)//7)
(1)
kk= exp(sim(za Zt, )/T)
where sim(.,) is the cosine similarity function, exp(-) is the exponential function, Z; fX ), and r is a scaling parameter. The model parameter updated iteratively by gradient descent, i.e., 8 -VVol f), where 7) is learning rate.
Using the representations of all windows centered at each point in X, i.e. {z, & [1, }. a change metric De is derived for each point x, E X. which represents the extent that x; is boundary. For example, the change metric in TS-CP² employs the distance (i.e., cosine similarity) between the embeddings of adjacent points,
Change Metric and Detection:
Vt
Ze+1) MA MA(sim Ze+ 1)41))))
where MA calculates a simple central moving average and NORM is min-max normalization over all timestamps to rescale value between 0 and 1. Then, similar to binary classification, the points whose change metric exceeds a certain threshold yp are identified as boundaries,
Č = {t De 2 4 where t [1, TT}-
Goal: Obviously, an effective change metric is crucial to the success of boundary detection. Therefore, we propose novel change metrie, ","r". using the curvatures in the representation space instead of the consecutive distances in the representation space.
3.2 Curvature-Based Change Metric
A trajectory usually refers to the path or track that an object (e.g., human and vehicle) in motion follows through space and time [44]. Thus, get to Definition 3.1 if we think of object point floating in the representation space.
Definition 3.1 (TRAJECTORY). A representation trajectory (simply trajectory) T is curve specified by a sequence of representations at consecutive timestamps and denoted as T (2,) (z)).
The curvature at specific point a curve is the rate at which the direction of the curve changes instantaneously at the point [17] It well-defined concept in geometry and quantifies how sharply or gradually the curve bends or deviates from a straight line. We employ the definition designed for trajectory 45] For three timestamps in order, t_, 1, and tt, where -- < < t+. consider their representations 2,-, Z, and Z,+. Two difference vectors, Z, Zz- and Zt-
Ze Bg
Zt-
Figure 3: Turning angle.
Ze naturally derived, and the turning angle O, between
them in Figure iss calculated by
(z. 2,-) ) (z,+ Z,)
0, arccos
(4)
Zz Zi-1 -2+
Each value of 0g ranges between 0 and n, where E [2, |T|- the direction changes between the two difference vectors, i.e. how much a difference vector rotates per unit length, as defined in Definition 3.2
1]. Then, the curvature is the rate of
Definition 3.2 (CURVATURE). The curvature at timestamp t in a representation trajectory T is the turning angle 9 divided by the sum of the difference vector lengths,
0₂
K2
2
2, + 12+ zel
According to our observations and intuitions described in Section I the curvature of an intra-segment point is higher than that of an inter- seter-segment point. Thus, the curvature defined in Definition 3.2 can be used as a change metric. The computational complexity remains (d'T), consistent with that of TS-CP2 where cosine similarity is computed instead of curvature. For stability, the timestamps t- and + in Eq. (5) are determined to be w 2 timestamps before and after timestamp t, i.e., t = w and + + w. We set to 5% of the mean segment length, which observed work well in most situations. Please refer to Section 4.6]abou the sensitivity analysis on the value of w. Definition 3.3 concludes our novel survature-based change metric.
Definition 3.3 (CHANGR. METRIC). The curvature-based change metric at timestamp / becomes
Jt youre MA(1 NORM(K)));
(6)
where is obtained from Eq. 5 and MA and NORM are the same Eq.

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In Definition 3.3
we normalize the curvature to a scale of to 1. Subtracting this normalized curvature from ensures that the change metric increases as the curvature decreases while maintaining the scale. Finally, a moving average is employed to smooth the curvature and mitigate fluctuations.
3.3 Change Metric Thresholding
Once the change metric of in Eq. (6) is prepared, it is possible detect boundaries by finding the points where 2/t 2 sp, as formulated by Eq. (3) Therefore, it is necessary to develop a heuristic for determining the threshold yp, and additional information
be utilized for this purpose. Such additional information includes the mean segment length and the validation dataset. If the mean segment length, i.e., the average of the lengths of segments distinguished by boundaries, is known, the estimated number of boundaries can be calculated by dividing the total number of timestamps by the mean segment length. The threshold ip is then determined to obtain the estimated number of boundaries. Alternatively, if we have validation dataset, we select the threshold şo that yields the best performance based on an evaluation measure. Empirical evaluation in Section employs the mean segment length in thresholding.
3.4 Theoretical Analysis
Our theoretical analysis is conducted by show- ing the following properties: (1) the intra- segment points in the representation space are confined within a smaller hypersphere than the inter-segment points, as shown in Figure (2) the mean curvature of a representation trajectory increases as the radius of the confining hyper- sphere decreases, which leads to the rationale behind Definition 3.3
Boundary
Enclosing ball
0
inter
Inter
R Rinter
0 intes
0 intras
Oc:
unto 0 entra
'
Riba
Definition 3.4 (CONFINEMENT), Consider subsequence of specific class C, Xc. (xe (x)tttattris 1 as well as its representation trajec- tory, Tc, = (21))=nc in Figure Then, To, is confined within a hypersphere Sc, Rdd centered Oc, E Rdd of radius Rintra if and only if [zz - 0c, < Rintra holds for all t E [[tstart fend].
Class G confining bell
Class Cj confiningball
Figure 4: Comparison of the curvatures between intra- and inter-segment points.
Definition 3.4 comes from widely known fact that representation (contrastive) learning produces class-sepane elass-separated representations 18 19). The augmented positive examples form a connected graph based on augmentation overlap; thus, the alignment of positive examples by contrastive learning will cluster the examples of the same class together and lead to class class- separated representations [8].
Proposition 3.5 (CONFINEMENT RADIUS). Consider transition from class C₃ to another class Cj in Figure 4 Let Sc, gdd be the confining hyperspheres for C₁ and Cj. respectively, of radius Rintra- Then, consider larger hypersphere of radius Rinter that encloses the inter-segment points (in red) as well as Sc, and Sc,, Thus, Rintra Rinter holds by definition.
Based on temporal coherence 7 13 46
inherent in time series, we make an assumption on the representation trajectory before proceeding to the second step.
Assumption 3.6 (EQUILATERAL RANDOM WALK), A representation trajectory T = =02+) is Markov chain, where z, is sampled over the surface of the unit hypersphere centered at 2 and also contained in a confining hypersphere of radius R. That is, [zz 2,- = 1 (t E [2, [T]] and
||z.|| < [1, [T]]] such that R >
Under Assumption 3.6
the curvature in Eq. (5) becomes the turning angle in Eq. (4) because the
denominator is reduced to a constant when = 1. Then, when a given representation trajectory T is confined by a hypersphere of radius R, its mean curvature is defined by
(R) T
Σ
(7)
Z,ET
where 12,,,18 [0.] is the expectation of the curvature at timestamp /. with respect to the distribution of the representations in the confining hypersphere of radius R.
5

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Lemma 3.7 (MEAN CURVATURE). Consider representation trajectory T confined in hypersphere Then, the mean curvature (R) decreasing function of the
of radius R under Assumption 3.6 radius R, ie., aR (R) < 0.
The proof of Lemma 3.7 is provided by Diao et al. 47 The mean curvature is rigorously formulated as complicated integral. By a simulation of random walk with one million steps. the decrease in the curvature represented by the linear function 3.53 1.21/t and the function 7 0.65/ R2-5 for two different regimes of R.
The representation trajectories confined within the hyperspheres of radii Rantra and Ranter in Figure 3 are called intra-segment and inter-segment trajectories as well as denoted by Tintra and Tinter, respectively.
Notation.
Putting Proposition 3.5 and Lemma 3.7 together, the observation on the difference in the curvature
finally formalized by Theorem 3.8
Theorem 3.8 (CURVATURE DIFFERENCE). The mean curvature of an intra-segment trajectory is greater than that of an inter-segment trajectory, iee, KTinee (Rintra) > KTintor (Rinter)-
Proof. Because Rintra < Rinter by Proposition 3.5 KTenta (Rintra) > KiFentor "iinttorRiittee (Rinter) obviously holds by the decreasing nature of KC(R) of Lemma 3.7
Theorem 3.8 can be intuitively explained if we consider the special case in which the next repre- sentation of Zintra or Zinter lies on the surface of hypersphere, as visualized in Figure [ Since a smaller radius necessitates sharper turn, fintra >> Ointer holds true. In this particular instance, where Zintra or Zinter is an orthogonal projection onto the surface, the turning angle can be expressed as π - arccos, 1 decreasing function of R. Please refer to Appendix A for more details.
2R*
3.5 Empirical Validation
The findings in the thcoretical analysis also align well with the visualizations of the representations from a real dataset. displays three representation trajectories in the representation space of two principal components, which obtained by the TPC method with d' = 32 for the mHealth dataset. Each representation trajectory includes 100 points centered at a boundary, where each point is sampled every ten points in the original trajectory. Inter-segment points within five sampled timestamps from the boundary are denoted by "x"', while intra-segment points are denoted by "o" The color of each symbol indicates the value of our change metric-i.e., I-curvature. Obviously, inter-segment points have higher values of the change metric than intra-segment points. Interestingly, in Figure []] the distance between two consecutive representations remains similar regardless of whether they are intra- or inter-segment points. This result reaffirms the existence of temporal coherence the representation space, which could reduce the accuracy of distance-based methods. Moreover, it is evident that the representation trajectories of intra-segment points exhibit clearer confinement, resulting in more closed shapes and larger average turning angles. The representation trajectories of inter- neter-segment points have fewer rotations and produce relatively straighter shape.
Low
Change metric
High
:nnra
Inter
3
0
-2
-1
-4
-5
-4
-7.7
-6
4
-2
4
-2
5
7
(a) Boundary at f. (Stand
247399.
(b) Boundary at f 321 128..8 (Walk-tBend)
(c) Boundary at = 339047. (Run-+Jump)
Figure 5: Three representation trajectories in the space of two principal components in mHealth.

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4 Evaluation
4.1 Experiment Setting
Datasets:
The profiles of the four datasets used in our experiments are summarized in Table which lists the number of timestamps, mean segment length, number of classes, data dimen- sionality, sampling rate in Hz. and
Table I: Summary of datasets and hyper- parameters.
Dataset
Timestamps
Length
Class
Rate
#Bo
Window
Epoch
WISDM
343002
697
20
491
HAPT
407807
903
50
450
100
50
mHealh
343195
2932
50
50salads
496250
551
19
2048
30
898
50
100
number of boundaries. WISDM 488.
HAPT 49]. and mHealth 49) are human action recognition datasets, which are measured by single or multiple accelerometers and/or gyroscopes. 50salads [50] is video dataset that captures 25 people preparing salads; the I3D features of 2048 dimensions are extracted, following Farha and Gall [3. The set of ground-truth boundaries, C, defined as the set of the timestamps where the class changes. The dimensionality of the representation space is set to d' = 8 for WISDM and HAPT and d' = 32 for mHealth and 50salads, considering their data dimensionality.
RECURVE Details: To obtain the point representations, we employ two time-series represen- tation learning methods, TPC proposed in TS-CP² [3] and TNC 14. RECURVE+TPC and RECURVE+TNC indicate the two implementations depending on the representation learning method. temporal convolutional network (TCN) trained in both methods. Note that any representation learning method can be combined with RECURVE. The window size, 21, and the number of training epochs for each dataset are shown in Table [ where the window size approximately twice the sampling rate. The learning rate is set to 0.005 for all datasets. The hyperparameter w, indicating the length of a representation vector, is set to 5% of the mean segment length. After obtaining a change metric for each timestamp, in Eq. 6 the same normalization is applied to all windows from an individual dataset. The moving average in Eq. (6) is computed using the ten timestamps preceding and following each timestamp. RECURVE is implemented using PyTorch 1.13.0, and its source code is available at htttps://gitiiii
Compared Methods: RuLSIF 24, KL-CPD 26 and TS-CP* I3] are chosen as the representative
method from each of the three categories in Section 2.1 The window size in Table [ applied to all compared methods for fair comparison. A multilayer perceptron is used for the regressor of RuISIF. The hyperparameters of RuI.SIF and KI,-CPD are favorably determined by a grid search, as detailed in Appendix B The public implementations of SuI and KL-CPI-T are used for our experiments. TS-CP" is the closest to our work, and its main mechanism is briefly described in Section 3.1. Because representation learning itself is shared between TS-CP" and RECURVE when TPC is used, the same hyperparameter setting is applied to both methods whenever possible. TS-CP2 is re-implemented using Py PyTorch 1.13.0 for direct comparison with RECURVE.
First, the Area Under the ROC Curve (AUC) is measured by considering boundary detection as binary classification with a binary label vector E (0. converted from C. Following Deldari et [13, an error margin introduced to accommodate some noise from annotation and detection. A detected boundary is considered to be correct if it lies within timestamps from one of the ground-truft -ruth boundaries. For this purpose, y relaxed to
Evaluation Measures:
1 if p S to + p where ty E C 0 otherwise.
Vt
(
Then, for t E [[1, TJ. whether (Út in Eqs. 2 or vp is compared against ys in Eq. We use multiple error margins, p E (5, 10, 20). since a margin could be different for diverse applications M. Second, the mean LOCation distance (LOC) is measured, which is the average distance from detected boundary to its closest ground-truth boundary 22 51. The LOC measure useful for checking the preciseness of the boundaries detected.
Regarding the threshold ip, the AUC measure does not require a specific value because it evaluates the true positive and false positive rates over given range. For the LOC measure, two values are used for each experiment: one determined to achieve the best F1 score, and the other is determined
7

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by the heuristic based on the mean segment length in Section 3.3 where the estimated number of boundaries multiplied by p 10, taking the error margin into account.
For each evaluation measure, we conduct every experiment five times with different seeds and report the average well the standard deviation. We use Intel(R) Xeon(R) Gold 6226R CPU @ 2.90GHz and NVIDIA RTX 3090 for every experiment.
4.2 Comparison with State-of-the-Art Methods
Tables and 3 display the AUC and LOC measures for the five methods across the four datasets. The AUC measure is presented in Table 2 with varying the error margin p. RE- CURVE outperforms the other bound- ary detection methods, where the opti- mal representation approach varies for each dataset. RECURVE wins against TS-CP2 in all datasets, irrespective of the evaluation measure. This find- ing demonstrates that the curvature more effective for boundary detec- tion in temporally coherent time se- ries where the class changes gradually. WISDM, HAPT, and mHealth exhibit periodicity in certain classes, includ- ing Walk and Run. This periodicity would produce closed shape for intra-segment trajectories and
Table 2: Overall detection accuracy in the AUC measure (the best results in bold).
Methods
WISDM
НАРТ
mHealth
SUsalads
5
0.559-00.05
0,797-00.001
0.598+00.002
0.606-0.006
RuLSIF
0.560=00005
0.797=0.001
0.599=0.002
0.00=00.038
20
0.563-+0.005
0.797+0.001
0.600+0.002
0611
0.697=0.000
0.868=0.003
0.842:0.117
0.682-0.083
KL-CPD
10
0,702=0.000
0,873=0.004
0.849=0.115
0.88-00.03
0.710+0.000
0.875+0.005
0.856+0.100
0.689
5
0.815=00.012
0.692
0.560=0.014
0.600-0.010
CP2
0.820.10.012
0.69510.006
0.682_0.009
20
0.823=000013
0.697=0.006
0.561#0.010
0.685=0.008
0.897=0.003
0.909=0.001
0.95420.003
0.719_0.005
RECURVE
10
0.901=0.004
0.913=0.001
0.954=0.003
0.723-0.006
+TPC
0.902-0.003 20
0.919
0.955+00000
0.729 0.006
RECURVE
5
0.880=0.0044
0.863=0.017
0.979:0.004
0.394-0.016
+TNC
10
0.889=0.004
0,867=0.017
0.980==0004
0.595=0.116
20
0.905=000044
0.876:20.017
0.980=0.005
0.600=0.015
increase their curvatures in the representation space, enhancing performance of RECURVE. In particular, when 20, RECURVE outperforms the second-best method by up to 12.7% in terms of the AUC measure for the mHealth dataset.
Table 3: Overall detection accuracy in the LOC measure (the best results in bold).
Methods
LOC (thresholding by best F1)
LOC
| (threshlddng by mean segment longth)
WISDM
HAPT
mHHealth
50salads
WISDM
HAPT
mHHealth
50salads
RaL.SIP
420
108.2:0.11
780
184 A63
429.
156.0=0.92
802.6 30.18
189.24
KL-CPD
12.20
121.5
306.41126.5
179.513.853
198. 198.3.1..52 329
113.0122.445
352.6. 119.7
176.6110017
TS-CP*
166.6+ 7.80
386.6+331.04
879-4-62.57
119.0+6.712
183.1+15.13
404.2+:3.60
423.8-44.39
12944+5001
RECURVE+TTPC
13.25111.90
178.4136.05
341.0247.93
93.76.1.475
RECURVE.:INC
2100+-122.3
47.92-#2.864
224.0-211.2
175.0422638
219.8+102.2
50.71+1.559
239.6-212.4
178.8+2087
4.3 Detailed Investigation on Change Metric Quality
We display the average values of the change metrics separately for each pair of classes using the HAPT dataset, which was chosen for ease of visualization due to its small number of classes. Figure 6a depicss the inter-class embedding distance, which determined by the Fuclidean distance between the centroids of point representations of given classes. The values of the change metrics are averaged
Lie Stnnd Sit Down Up Walk
15
Lie Stand Sa Down Up Walk
1.00
Up Walk
1.00
Liee Stand Sit Down
10
-0.5
0,900 Walk Up Down Si Sland Lie
-0.90
Walk Up Down Sit Stand Lic
Up Down
(a) Inter-class embedding distance.
(b) Averaged gllll (TS-CP²).
(c) Averaged "F (RECURVE).
Figure 6: Heatmaps of the inter-class distances and values of the change metrics between the classes in the HAPT dataset. A gray box indicates no transition between two classes.

---

across the inter-segment points for each distinct class transition. Figures 6b and 6c are obtained by the distance-based change metric b of TS-CP2 and the survature-based change metric Ut of RECURVE, respectively. Intriguingly, Di generates high values for all class pairs in Figure 6c which indeed explains the overall high accuracy in Tables 2 and In contrast, Figure 6b y/ only generates high values when the inter-class embedding distance is sufficiently large (i.e., abrupt change), whereas it generates moderate values when the inter-class embedding distance is small (i.e., gradual change). That is, Figures (6a and [bb show a very high correlation. In summary, 22 insensitive to the degree changes whereas yt is not. Therefore, this result demonstrates the superiority of the curvature-basee change metric over the distance- basee change metric.
Figure 7 magnifies six class pairs selected from all class pairs depicted in Figures 6b and 66 For example, Stand=tii and
TS-CP=
RECURVE
25
Lie->Stand are accompanied by rapid body movement, and both TS-CP2 and RECURVE capture the boundaries well, evidenced by the high density the interval close to I. In contrast, when two action classes are com- parable, as in Stand-Walk, Down-Up. and Walk->Down, the values of the change metric of TS-CP2 disperse to other intervals, result- ing in decrease in detection performance. RECURVE maintains the same shape in all density plots due the remarkable effective- ness of our curvature-based change metric.
Density
25
0.8
1 0.8
0.8
0.8
0.8 1 0.8
Change metric
Stand Lie sit stand Walk Stand
Stand Up Down Walk
Up Down
Figure 7: Distribution of the change metrics for each class transition in the HAPT dataset.
4.4 Evaluation Respective to Gradual and Abrupt Changes
Table shows the corresponding AUC scores for gradual and abrupt changes in relation to various distance-based metrics. If the inter-class embedding distance between two classes is shorter than certain threshold, the change between the two classes is categorized as gradual. Here, the threshold established such that gradual changes represent 20% of all changes. Abrupt changes are excluded when measuring the improvement for gradual changes, and vice versa. The distance-based metrics, denoted by DISTANCE, are further categorized depending on whether the Euclidean distance or the cosine similarity is used. Note that RECURVE adopts the curvature- based metric.
Overall, RECURVE demonstrates greater improvements for gradual changes (AUC-Gradua) than for abrupt changes (AUC-Abrupt). For example, in the HAPT dataset with p 5 and the TPC representation, the increase in the AUC measure from TS-CP2 RECURVE for gradual changes is 32%, which is significantly higher than the 21 % increase for abrupt changes. This result confirms that the curvature-based method is particularly effective for detecting gradual changes because captures subtle variations in the data that are not readily discernible through distance-based methods. In conclusion, RECURVE is versatile to support both gradual and abrupt changes.
Table 4: Detection accuracy in the AUC measure respective to gradual and abrupt changes.
Methods
Repr.
2
WISDM
[APT
AUCC-mmdull
miliaath
WISDM
ПAPT
AUC-Ahrupt milealth
50calads
3
0.16660.229
0.510+0.005
0,44400.099
0.716-6.010
0.696+0.010
TPC
10
0.600.0.000
0.50010.029
0.22110.005
0.58910.005
0.695.0.009
0.71710:010
0.69910.010
0624100027
DISTANCE
20
0.091:0.009
0827200025
0.52150.005
0.0051000005
0.0905000008
0.718:0:000
0.700:01010
(Fuclidean)
0.71550.012
0.02200:022
07706900017
0356=+0.057
0.720=00:008
0.546-0.007
0.600-0:011
TNC
10 0.724100014
0.608100022
0.70810.017
0.558:0.056
0.73510.000
0.849:0.007
0.84710.025
063210011
0.7341000117 20
0.008:0:022
0,00910.017
0.50110:056
0.75420.011
0.841200077
0.319:1..029
0.33610.011
DISTANCE (Cosine)
(=T5- (
TPC
0.81410.015 20
10
0.811100.014
0.807=000014
0.006:00:223
061310:022
0.602-0.023
0.54810.014
5.01100.111
0.640000115
0.67410.010
06711000010
0677.10.09
0.84510.010
0.838+0.009
0.84710.010
0.34200:021
0.746:00021
0.747L00020
0030--0.339 0.61-
0.66610.011
063710.010
0.70610.005
0.71410.037
0.77910.013
0.77420.013
0.78910.017
0.59110.022
0.79410.012
0.905-9.006
0.89210.021
0.681100011
TNC
10
0.7877-0.012
0.783=0:012
0803+-0.015
0 0.502=0:4122
01 11800-0.011
0.546-0.044
0500+
0.65-00.010
0.81410.013 20
0.79910.014
0.04600.009
0600100012
0 0.84100.011
0.50100.004
0.281000117
0.801000000
0.888-0.004
0.886:0.0099
0.712=1.006 0.939:50000
0.923=000015
0.945=0.002
0.9554000344
0.729=0.055
TPC
0.893:0.005
0.891:0.009
0.94010.000
0.715:00006
0.927:0.001
0.73
RECURVE
0.94100000
0.8%600.009
0.94110.000
0.722.0..96
0.927100.012
0.9881000044
0.734100011
0.867501.00
0.773:0:035
0.975100.003
0.551=00005
000000
0.531
0.988:00.06
0.600
TNC
0.875+00000
0.779-00.038
0.977:0.005
0.553+000034
0.910=0.006
0.933-0.007
0.989--0.086
0.00--0.016
0.89010.0008
0.79110.033
0.97710.004
0.55810.003
0.92710.000
0.92910.002
0.99010.006
0.606.10.05

---

4.5 Visual Analysis of the Change Metrics
Figure 8 visualizes the fluctuations of various change metrics obtained from the HAPT dataset using the TPC representation. TS-CP" fluctuates rapidly during the changes and seems have many false negatives at the rightmost boundary area. However, RECURVE indicates the inter-segment points much more clearly than TS-CP², without excessive false positives and false negatives. The reliable detection is achievable by taking into account both the turning angle (the numerator in Eq. ( and distance (the denominator Eq. (5) in the representation space.
1.00
TS-CP2
0.75
RECURVE
0.50
0.25
0.00
1000
2000
3000
4000
5000
Timestamps
Figure 8: Change metric scores from the HAPT dataset with the default configuration. A gray-shaded area represents the inter-segment points between two class segments.
4.6
Sensitivity Analysis on the Hyperparameter
Table (oole:5sseee 5|sows the performance of RECURVE while varying the hyperparameter (see Definition
3.3) when the error margin y for the AUC measure is fixed at 10. The value of w: ranges from 0.25x 4.00x of the default value, which set to 5% of the mean segment length (indicated by 1.00x) If the value of w were too large, the denominator of Eq. ( would be too large for any point in time series, and the curvature would unable distinguish between intra- and inter-segment points. If the value of w were too small, some noise in point representations would distort the curvature. Under this trade-off, the default value performs the best in terms of the AUC measure when it is averaged over the four datasets and the two representation learning methods. On dataset with lengthy segments, such as mHealth, the sensitivity tends to decrease, and there is small variation when varying the value of w.
Table Performance of RECURVE with varying the hyperparameter w (the best results in bold).
AUC t
(chresholding by meun segment length)
Repr.
WISDM
HAРT
mHealth
50salads
WISDM
HAPT
mHHalhh
50salads
0.25
0.832=0.015
0.901=0.004
0.911=0.000
0.685=0.0007
358.7=89.00
40.41=22235
654.7433.00
136.9+2..064
0.89110.006
0.91410.002
0.95310.004
0.703.0.005
239.31139.6
37.4221.309
538.9149.69
120.012.127
TPC
1.00x
0,901 0.901:00.044
0.913:40.001
0.954
0.723:000066
178.4:36.05
34.28:00727
341.0:47.99
93.76::7.475
2.00x
0.89210.003
0.887100.00
0.92710.003
0.692.1.000
252.9197.20
42,74:15.383
821.21153.00
94.2116.153
4.00x
0,861+0.000
0.847++0.002
0.893+0..004
273.8-119..0
53.00+10.77
628.24-39.66
0.25x
0.824+-0.016
0.842:-0.011
0.956=00.09
0.580=0.0155
249.7=337.59
52.95+33.698
213.74-110.2
222.0+-7.445
0.869:0.009
0.850:20.012
0.978=00005
0.587=0.014
231.3::79.25
21.15+2..85
2364::136.5
218.9+16.11
TNC
1.00x
0.889+00.004
0.867-0.017
0.980-0.004
0.595-+0.016
219.8-102.2
1.599
239.64-212.4
178.8+-20.8 20.87
2.00x
0.897=0.002
0.827=0.019
0.962=0.007
0.583-20.000
196.0=79999
58. 1,58,
346.6+30599
179. 4-
0.87110.002
0.77310.016
0.568.1.0009
97.3816.328
265.61192.00
183.2111..44
The sensitivity analysis on the representation dimensionality d' is available in Appendix
5 Conclusion
In this paper, we present RECURVE, a novel boundary detection method that uses the curvature of representation trajectory to replace the consecutive distance for a change metric. Theoretically, the mean curvature an intra- antra-segment trajectory greater than that of an inter-segment trajectory due to the confining nature of the representations of the points within a single class. Unlike the consecutive distance, this property of the curvature is insensitive to the degree of the changes between two classes (segments). Our comprehensive experiments confirm that RECURVE. achieves up to 12.7% higher detection accuracy than state-of-the-ar methods. Overall, we believe that our work pioncers new direction for boundary detection in time series.
10

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Acknowledgments and Disclosure of Funding
This work was supported by Institute of Information Communications Technology Planning & Eval- uation (IITP) grant funded by the Korea government (MSIT) (No. 2020-0-00862 / RS- DB4DL: High-Usability and Performance In-Memory Distributed DBMS for Deep Learning, 50% and No. 2022-0-00157 RS RS-2022-112201135 Robust, Fair, Extensible Data-Cent Centric Continual Learn- ing, 50%).
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A The Proof of Theorem 3.8 in a Special Case
To explain Theorem 3.8 in a more intuitive way, we assume a special case in the 2-dimensional space as shown in Figure 4 in a 2-dimensional representation space. In the figure, there are three black dots in a row for the intra-segment trajectory (gray line) and the inter-segment trajectory (red line) respectively these three points in each trajectory. Here, 2; is an arbitrary point on the border of a confining circle Sc; the enclosing circle. 2,- is the point on the line between the origin and 2,- Finally, we consider Z,+ as a random point distributed uniformly on the unit circle centered at 2, while the locations of Ze and z are fixed. In this case, we compare the average curvature at the single timestamp of the intra-segment trajectory and that of the inter-segment trajectory.
Let have random walk composed of three points z,-, 2,, and Z
visualize
Lemma A.1 (AVERAGE CURVATURE). When the two points z, and 2₁ are fixed, the curvature k. at Z₁ averaged over the location of z, is
1
1
(R)
arccos 2R
(9)
Proof. When the confinement radius is R, the range of the turning angle is ia arccos 1 *] using the cosine rule on the triangle composed of the points of intersection between Sc, and the unit circle centered Zz.
Lemma A.2 (CHARACTERISTIC OF AVERAGE CURVATURE). When R>> 1, (R) decreases as the radius increases, i.e., KT (R) a decreasing function of R.
Proof. The derivative of ki an = = 2RVER=-T and the derivative is always negative when R > 1. Therefore, KT is decreasing function of R.
Similar to Theorem 3.8 Proposition 3.5 and Lemma A.2 lead to Theorem
Theorem A.3 (AVERAGE CURVATURE DIFFERENCE). The average curvature of an intra-segment trajectory KTntea (Rintra) is greater than that of an inter-segment trajectory KTonee (Rinter).
Proof. The radius of the confining circle of an inter-segment trajectory is always bigger than that of an intra-segment trajectory because an inter-segment trajectory traverses two distinctive class balls. As Ny (R) decreasing function, NTeener (Rantra) > Winser (Rinter).
Theorem A.3 concludes that the average curvature at 2; is greater if z; is at the border of a smaller confining circle. This result can be extended Theorem [38 intuitively. probability of a trajectory hitting the border decreases as a random walk can traverse in larger space. Therefore, the mean curvature decreases if a trajectory resides in a smaller confining circle.
R increases, the
B Hyperparameters for Compared Methods
For RuLSIF, we conduct a grid search for the learning rate (LR) = (0.05, 0.1, 0.2). the weight of L2 normalization >1.2 {0.01, 0.05, 0.1), and the parameter of the RuLSIF loss {0.00,, 0.05, 0.1}. When applying RuF to the four datasets, we use a multilayer perceptron with a single hidden layer with 100 units and train it with batch size of 32 for 50 epochs. For KL-CPD, conduct grid search to determine the optimal hidden dimensionality h = {10, 50, 100) of the RNN encoder/decoder, as well as the values for the hyperparameters λAE = {0.1, 0.01, 0.001 and AReal = (0.1, 0.01, 0.0013 which govern the influence of the reconstruction loss and the MMD2 loss on real datasets. For training the generator of KL-CPD, the batch size to 64, the number of epochs is to 3, and the learning rate is set to 0.001. Table o provides a summary of the determined hyperparameter values.
C Sensitivity Analysis on Representation Dimensionality
Table 7 shows the performance of RECURVE while varying the representation dimensionality d' when the error margin p for the AUC measure is fixed 10. The value of d' ranges from 0.25x to 4. x.00x of the default value, which is 8 for WISDM and HAPT or 32 for mHealth and 50salads
15

---

Table 6: Hyperparameter values of RuLSIF (left half) and KL-CPD (right half) after a grid search.
Dataset
L.
Aua
AxE
AReal
#hadden
WISDM
0.05
0.1
0.01
0.01
0.001
10
НАРТ
0.2
0.01
0.01
0.01
0.1
10
mHcalth
0.2
0.1
0.01
0.01
0.01
10
50salads
0:05
0.01
0.05
0.1
0.01
50
(indicated by x.00x). A trade-off point in the representation dimensionality exists for nearly all datasets. A representation space with an excessively high dimensionality is susceptible to the curse of dimensionality. If the value of d' were too large, the turning angle and distance in Eq. (5) would be indistinguishable across all timestamps would become nearly orthogonal and their distance would always be similar. If the value of to were too small, low-quality features would be extracted from the original time series by representation learning; thus, the performance degrades with an insufficient dimensionality as shown in the result of 50salads whose data dimensionality is 2048. Overall, the default setting suitable for achieving competitive performance for all datasets.
time series, as any two points in high-dimensional space
Table 7: Performance of RECURVE with varying the hyperparameter d' (the best results in bold).
Repr.
AUC
LOC
(chresbolding by mean sogmeat length)
WISDM
HAPT
mHHaalhh
50salads
WISDM
HAPT
mHHealhh
50salads
0.25x
0,870=00.007
0.842:0.001
0.942:0.007
0.719:0.006
349:9::32.99
307.0
553.41699770
97.64-16.153
0.50x
0.906:00.003
0.836+0.10
0.949=0.006
0.719=0.0077
377.5=443.7
104.4=130.3
586.14:39.80
97.33++5555
TPC
1.00x
0.901+00.004
0.913-00001
0.954=0.003
0.723 0.006
1784--3605
34.28-0.722
3410+44793
93.76++7.475
0.882:00001
0.905=0.003
0.942:0.005
0.718=0.005
180.6-59.32
35.62222669
600.9436882
100.9+3.814
4.00k
0.85710.001
0.90010.004
0.93720.005
0.71910.007
200.41:55.87
38.27
592.31:64.25
100.0166432
0.25
0,838:100046
0.86210.006
0.963-0.013
0.56110.018
168.1:79.54
50.371:4.293
241.0130..
198.7120.
0.50x
0.882+0.000
0.859=0.011
0.971=0.000 007
0.570=0.013
149.5=35.37
49.17=1.943
245.0=12250
198. 798.7337,33
TNC
1,00x
0.889
0.867110.01
0.980
0.595.10.016
219.2:102.2
$0.71=1.589
2396.1212.4
178.8.10087
2.00x
0.877:00.006
0.875:-0.009
0.972+0.003
0.581+0.011
290.4 148.8
55.35++1.00 1,800
260.7+-51.26
215.6.t 15.55
4.00x
0.8801000003
0.887 0.004
0.973
0.607
257.9
51.7241.010
280.5199.19
179.3.11.75
D Limitations
One notable limitation of RECURVE is the potential occurrence of false positives, particularly when dealing with short segment lengths that fall below the predefined threshold, denoted as w. For instance, consider a scenario where there is a rapid transition from one activity class to another, such as transitioning from walking for an extended period a brief sprint, followed by resuming walking. In such cases, if the segment length is shorter than the specified w duration, every timestamp within the brief sprint segment might erroneously be identified as a boundary, leading to a number of false positives. To mitigate this issue, we have implemented a strategy where w is determined as 5% of the mean segment length. However, is worth noting that this approach may encounter challenges in rare cases where the segment lengths exhibit significant variance. We will further investigate how segment length variance impacts the efficacy of RECURVE as future work.
16

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The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
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Reviews:
[{"review_id": "pJNXYWdjI3", "reviewer": "Reviewer_EoQb", "summary": "The paper introduces a boundary detection method called RECURVE for time series data, which leverages the curvature of representation trajectories as a novel change metric to accommodate both gradual and abrupt changes. Experiments on diverse real-world datasets demonstrate that RECURVE outperforms state-of-the-art methods, achieving up to a 12.7% improvement in detection accuracy.", "questions": "See weakness. I'd be happy to raise my score if my questions were answered well.", "limitations": "Yes.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Innovation. The paper introduces an innovative boundary detection method that identifies boundary points by analyzing the curvature of representation trajectories in time series data, offering a new perspective to the field.\n\n2. Intuitiveness: The authors intuitively demonstrate their points through Figures 1 and 2, which serve to elucidate the authors' insights and are persuasive.\n\n3. Theoretical Support: The paper substantiate the correctness of the proposed viewpoint through rigorous theoretical analysis, providing a solid theoretical foundation for the method.\n\n4. Experimental Validation: Extensive experimental design and results indicate that the proposed new metric is effective and well supports the paper's claims.\n\n5. Writing Quality: The writing is clear, logically coherent, and easy for readers to understand.", "weaknesses": "1. While the concept introduced in this paper is highly innovative—it detects boundary points from the perspective of representation trajectories— the intrinsic advantages of using a curvature metric have not been fully articulated: \n\n- 1.1 Although the paper proposes a method based on the trajectory perspective, it appears to rely primarily on data from three points in the trajectory (t+1, t, t-1), which leads to a final metric design that is similar to distance-based methods. \n\n- 1.2 Looking at Equation 3.2, there is a certain correlation between the curvature-based method and the distance-based method: k_t is negatively correlated with distance. This raises the question of whether the curvature-based method might actually subsume the distance-based method. Is the introduction of the turning angle theta truly necessary? The authors should delve into what the RECURVE method solves that distance-based methods cannot. \n\n- 1.3 Why is the curvature of intra-class points so large? Is this due to an excessively large theta or a too-short distance? Similarly, why is the curvature of inter-class points so small? Is this due to a too-small theta or a too-long distance?\n\n2. Assumption 3.6 raises some concerns. In this assumption, the authors posit that ||z_t − z_{t−1}|| equals 1. However, this assumption appears to be difficult to uphold in boundary point detection tasks; otherwise, distance-based methods should be universally ineffective. If this assumption does not hold, what are the implications for the perspectives expressed in this paper?\n\n3. Minor Question: The definition of Mean Total Curvature is somewhat perplexing; could you clarify why it is termed \"Mean Total\" rather than simply \"Mean\"?\n\n4. Other Inquiry: I have some queries regarding the Case Study. In Figure 5(c), I have manually marked five points: a, b, c, d, and e (you can view this through [this link](https://anonymous.4open.science/r/temp-AC2C/5c.png)). The curvature at point b, denoted as k_b=\\theta_b/(ab+bc), and at point d, k_d=\\theta_d/(cd+de). Clearly, \\theta_b is an obtuse angle and \\theta_d is an acute angle, implying \\theta_b > \\theta_d. Additionally, visually assessing the distances on the image suggests that (ab+bc)<(cd+de). Consequently, it would be expected that k_b>k_d. Considering that a smaller k indicates an inter-segment point, why does Figure 5(c) depicts b as an inter-segment point and d as an intra-segment point. Am I misunderstanding something here?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you so much for acknowledging the innovation of our new boundary detection method. We really hope that our responses are satisfactory to you.**\n\n---\n\n**`W1.1`** *Similarity to the distance-based metrics.* \n\nThe three points used for calculating the curvature are $t-w$, $t$, and $t+w$. Here, $w$ was introduced for stability. Then, the **simplified trajectory** (i.e., line segment) between $t-w$ and $t$ and that between $t$ and $t+w$ are considered to measure the turning angle at $t$. Thus, we believe that the trajectory perspective is incorporated into the design of our metric. \n\n---\n\n**`W1.2`** *Unclear necessity of the turning angle.*\n\nThank you very much for your insightful comment. Yes, you are right. The curvature seeks to maximize the **combined effect** of the turning angle $\\theta$ and the distance. In fact, Definition 3.2 is similar to the conventional curvature estimation in the geometry field [17]. **The turning angle is truly necessary especially for gradual changes.** Please see Figures 6 and 7. Between similar actions, e.g., from walking to descending (Walk$\\rightarrow$Down), the embedding distances across these two classes are small (Figure 6(a)). Consequently, the distance-based scores become low for these transitions because the embedding distances between consecutive inter-segment points should be also small (Figure 6(b) and the top row of Figure 7). Nevertheless, these drawbacks are fully cured by adding the turning angle to the change metric (Figure 6\\(c) and the bottom row of Figure 7). **Overall, Figures 6 and 7 clearly show how crucial the turning angle is to the change metric.**\n\n[17] Curvature and torsion estimators based on parametric curve fitting. Computers & Graphics, 29(5):641–655, 2005.\n\n---\n\n**`W1.3`** *Roles of the turning angle and the distance.*\n\nThe relative contribution of the turning angle $\\theta$ and the distance varies depending on the scenario. For gradual changes such as the transition from walking to descending in Figure 6, the score increases primarily due to a relatively large $\\theta$. In contrast, for abrupt changes such as the shift from walking to standing, the distance term has a greater impact on the score. In order to substantiate our assertion, we present the values of $\\theta$ alone in **Figure 9(d)** (refer to the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf)). Here, **$\\theta$ is greater for the transitions where the curvature-based method succeeds but the distance-based method fails**---i.e., for the transitions with smaller inter-class embedding distances.\n\n---\n\n**`W2`** *Implication of Assumption 3.6.*\n\nThis assumption is made in order to **concentrate on the effect of the turning angle especially for gradual changes**, where the distances between consecutive points are similar within a **local** range due to the temporal coherence of time series. This intention will be further elucidated in the final draft. That is, Theorem 3.8 is presented to prove that, **because of the effect of the turning angle**, the curvature is larger for intra-segment points than for inter-segment points. \n\nWe acknowledge that the assumption is not always true for datasets from the real world. However, RECURVE continues to be reliable because it takes advantage of the combined effect of the turning angle and the distance. Moreover, we **empirically** confirm the validity of Theorem 3.8 on the real-world datasets **without the assumption**, as shown in Figure 7.\n\n---\n\n**`W3`** *Minor: Term \"mean total curvature\".*\n\nSorry about the perplexity in the term. We will surely correct the term as the *mean curvature*.\n\n---\n\n**`W4`** *Other Inquiry: Actual calculation of the curvature in Figure 5.*\n\nWe are really impressed by your thorough review. Definition 3.2 and Line 165 of our draft state that points spaced $w$ apart are used in the curvature calculation rather than just consecutive points. This calculation makes the curvature insensitive to local noise and enables us to capture broader changes in the trajectory. Your understanding is correct except this point. The discrepancy between actual curvature scores and your calculations is mainly due to dimensionality reduction, where a 32-dimensional representation space is converted to a 2-dimensional space. Thus, unfortunately, the precise location of each point representation may not be accurately depicted on Figure 5.\n\nAdditionally, we have removed some redundant points from each trajectory visualization, showing only one point out of every ten, to prevent the figure from becoming too crowded and to make the overall trajectory easier to understand. Our visualization may have resulted in some confusion, and we will surely try to eliminate this confusion.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "biJMJmbhwg", "reviewer": "Reviewer_ZR9k", "summary": "This paper proposes a novel boundary detection method based on the curvature of a representation trajectory. The feasibility of the proposed algorithm is analyzed intuitively and theoretically, and the proposed method is experimentally shown to have good performance.", "questions": "1. How many types of boundaries are there?\n2. Which boundary problems can the proposed method solve and which boundary problems cannot it solve?", "limitations": "Yes", "rating": 7, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed method is simple, but effective.\n2. The proposed method is shown to be effective intuitively, theoretically and experimentally.", "weaknesses": "It may be better to first classify the boundaries into several categories, and then discuss the related work and experimental comparison for these categories respectively.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you so much for acknowledging the effectiveness and intuitiveness of our new boundary detection method. We hope that you can support our work during the discussion period.** \n \n---\n \n**`W1`**, **`Q1`** *Categorization of the boundaries and evaluation with the categorization.*\n\n**(1) *Categorization***: Thank you for your valuable comment. In our current draft, we categorize boundaries into **gradual** and **abrupt** changes depending on the speed of change. Gradual changes (e.g., Walk$\\rightarrow$Down) involve slow transitions over time, which can be challenging for traditional distance-based metrics to detect. Abrupt changes (e.g., Stand$\\rightarrow$Sit), on the other hand, involve rapid transitions that are typically easier to identify due to significant differences between consecutive points.\n\nOne way to explicitly categorize the boundaries is to use the inter-class embedding distance, which is the Euclidean distance between the centroids of point representations of given classes. If the distance is below a certain threshold, the transition between the two classes is considered gradual; otherwise, it is considered abrupt.\n\n**(2) *Evaluation***: Using Figures 6 and 7 in Section 4.3, we have already analyzed our experimental results with respect to the categorization. This analysis enables us to demonstrate how our method, RECURVE, performs in different scenarios and highlights its effectiveness in detecting both types of changes. To further improve the structure and clarity of our paper, we will expand Related Work to include more references on gradual change point detection.\n\n---\n\n**`Q2`** *Capability of RECURVE according to the categorization.*\n\nPer your suggestion, we show the performance results separately for gradual and abrupt changes in Table 5 (see the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf)). It is obvious that **RECURVE can support both types of boundaries**, also as demonstrated in Section 4.3. Specifically, in the HAPT dataset with $p=5$, the increase in the AUC from TS-CP$^2$ to RECURVE+TPC for gradual changes is 32\\%, which is significantly higher than the 21\\% increase for abrupt changes. The findings presented in Table 5 indicate that RECURVE is notably more efficient in handling gradual changes. A tricky case with very short segments is discussed in Appendix D.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "unxfr4C2cd", "reviewer": "Reviewer_vKgw", "summary": "This paper proposed to use curvature as the metric to detect the boundaries. Empirical and experimental results also show that the confining box is different for inter- and intra- variable data points.", "questions": "- How is the sensitivity of the parameter analysed? Are the results presented with the test set or is it with the evaluation set? \n- In Figure 5 (b), the curvature of some continuous intra-variable data points seems to have a similar curvature (approximately a straight line) compared to the curvature between the variables from a different class. Can you clarify this observation and how it may affect the proposed method?\n- What is the computational cost of the proposed method compared to baselines?", "limitations": "The limitation is well stated in Section D.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The idea of using curvature for boundary detection is novel in time series forecasting.\n- The paper is well-written, figures are clear and informative, making the paper easy to follow.", "weaknesses": "- The paper lacks a strong foundational motivation for introducing the curvature-based boundary detection method. The rationale behind why curvature would be a better metric compared to existing methods is not convincingly argued. From Figure 5, it seems that the distance-based methods can also split (a) and (c) well while for (b), the curvature difference is also not very big.\n- Many related boundary detection methods are mentioned in the related work but not directly compared in the experiment. It would be beneficial to include a more comprehensive comparison to show the effectiveness of the proposed approach.\n- RECURVE is highly dependent on the quality of the learned representation which makes it also holds the same limitation as a distance-based model when the representation of a time series data is not properly learned.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you so much for acknowledging the novelty and intuitiveness of our new boundary detection method.**\n\n---\n\n**`W1`**, **`Q2`** *Lack of a strong foundational motivation and interpretation of Figure 5.*\n\n**(1)** We anticipate that our clarification will effectively address your concerns.\n* ***Rationale***: In Introduction, Figure 1 empirically shows that the distances between consecutive points largely overlap between intra-segment points and inter-segment points, especially for gradual changes (Up$\\leftrightarrow$Down). Figure 2 illustrates the foundational motivation of the curvature-based metric. As long as the point representations are confined into a class ball, the trajectory of intra-segment points make sharp turns more frequently than that of inter-segment points, regardless of gradual and abrupt changes. This benefit of the curvature is proven in Theorem 3.8. \n* ***Evaluation***: Furthermore, **Figures 6 and 7 clearly demonstrate the advantage of the curvature-based metric over the distance-based metric**. For gradual changes, e.g., from walking to descending (Walk$\\rightarrow$Down), the embedding distances across these two classes are small (Figure 6(a)). Consequently, the distance-based scores become low for these transitions because the embedding distances between consecutive inter-segment points should be also small (Figure 6(b) and the top row of Figure 7). In contrast, regardless of gradual and abrupt changes, the curvature-based scores are sufficiently high (Figure 6\\(c) and the bottom row of Figure 7).\n\n**(2)** **Figure 5 is obtained using two principle components out of 32 dimensions**. Thus, the exact position of each point representation may not be reflected on the 2-dimensional plot. The color of a point indicates the exact curvature score calculated in the 32-dimensional space. The purpose of Figure 5 is to support our motivation that the intra-segment points are somewhat clustered whereas the inter-segment points are not. In order to minimize confusion, we will identify the plots that most accurately represent the original positions in the 32-dimensional space.\n\nIn conclusion, we trust that the curvature-based metric's superiority over the distance-based metric is readily apparent to you. We will endeavor to further improve the presentation in the final draft.\n\n---\n\n**`W2`** *Comparison with more baseline methods.*\n\nIn fact, we choose the representative method from each of the three categories (statistics-based, pattern-based, and representation-based) mentioned in Related Work. Because the representation-based methods are regarded as the state of the art, we add another representation-based method, which is a variation of TS-CP$^2$ with the TNC representation learning technique. **RECURVE also beats this additional method**, and please see Table 5 in the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf).\n\n---\n\n**`W3`** *Dependency on the learned representation.*\n\nWe agree with you. However, as representation learning (or contrastive learning) for time series advances remarkably [a], it is unlikely that the quality of the learned representation is very poor. We will discuss this limitation in Appendix D of the final draft. It is crucial to note that, **even if the quality of the learned representation is sufficiently high, the distance-based method is susceptible to gradual changes whereas the curvature-based method is not**.\n\n[a] Universal Time-Series Representation Learning: A Survey, arXiv:2401.03717, 2024\n\n---\n\n**`Q1`** *Hyperparameter sensitivity analysis.*\n\nWe have conducted an analysis of the sensitivity of two hyperparameters $w$ and $d^\\prime$, as shown in Section 4.4 and Appendix C, respectively. The sensitivity analysis was done using the test set. It was confirmed that the default setting of the two hyperparameters is suitable for achieving competitive performance across for all datasets, and the sensitivity was not a concern. \n\n---\n\n**`Q3`** *Computational cost.*\n\nGiven a representation dimensionality $d^\\prime$ and a time-series length $T$, the computational complexity of curvature computation in RECURVE is $O(d^{\\prime}T)$. **This complexity is the same as that of TS-CP$^2$, thereby ensuring a comparable computational burden**. It is noteworthy that with a low representation dimensionality, i.e., $d^\\prime=32$ as the default, and a linear scaling with respect to the length of the time series, RECURVE demonstrates efficiency in the context of lengthy time series. The detailed cost analysis will be added in the final draft.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TX4YmfGDoY", "reviewer": "Reviewer_aPkZ", "summary": "The authors propose a novel boundary detection method for time series based on measuring the curvature of the local trajectory of a learned per-timepoint embedding/representation. Some theoretical justifications are provided for the proposed method. Empirically, it is shown to work better than a number of existing methods on a few time series datasets.", "questions": "I wonder whether one can go a step further and establish some notion of optimality under the right settings. Suppose that the representation for each class is drawn from a gaussian with class-dependent mean/covariance, then there exist optimal statistical tests for the inter-class change points. How would the proposed method perform compared to the optimal boundary detector?", "limitations": "N/A", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The proposed method seems novel and the empirical results are encouraging. The paper overall is easy to read and the main idea is easy to understand.", "weaknesses": "The whole idea builds on the fact that there exists a learned representation that more or less satisfies the locality assumption (e.g. intra-class points live within a hypersphere). Although some limited empirical evidence is presented, it is unclear in general whether this is a common phenomenon or something else needs to be done to encourage representations that more faithfully respect the assumption.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you so much for acknowledging the novelty, evaluation result, and intuitiveness of our new boundary detection method.**\n\n---\n\n**`W1`** *Locality assumption on the learned representations.*\n\nThank you for your insightful comment. Yes, RECURVE is built upon class-separated representations. **We believe that it is a widely known fact that representation (contrastive) learning produces class-separated representations [18, 19]**. According to [18], the augmented positive examples form a connected graph based on augmentation overlap; thus, the contrastive learning will cluster the examples of the same class together and lead to class-separated representations. We use two representation learning techniques, TPC and TNC, for our evaluation. Consecutive classes tend to be well separated on the embedding space, as shown in Figure 5. RECURVE is empirically shown to work well with these two representation learning techniques.\n\nAs a topic for future work, we will continue to explore the design of a customized representation learning technique for RECURVE. Your comment is indeed inspiring!\n\n[18] Chaos is a Ladder: A New Theoretical Understanding of Contrastive Learning via Augmentation Overlap, NeurIPS 2022\n\n[19] InfoNCE Loss Provably Learns Cluster-Preserving Representations, COLT 2023\n\n---\n\n**`Q1`** *Optimality under the right settings.*\n\nThis is a very good suggestion! The Gaussian mixture model can be applied to a set of point representations. Then, based on the probability distribution of a point, some statistical tests can be done to distinguish between intra-segment points and inter-segment points. It is difficult to implement this procedure during the rebuttal period because of the short time allotted, so we will include this comparison in the final draft. Given this setting, where point representations are well clustered, we anticipate that RECURVE will be very close to the optimal boundary detector.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "E8JzSF9RmH", "reviewer": "Reviewer_PiYP", "summary": "This paper proposes a novel approach called RECURVE (Representation trajEctory CURVaturE) for boundary/change point detection based on time series representations. RECURVE measures changes in representations of time series windows over time, based on curvatures instead of distances in the representation space like existing work. The intuition is that the direction of the representation trajectory tends to change more sharply at points within a segment than at points between segments. This is because representation learning aims to learn class-separated features, and intra-segment points are confined within a class-specific ball, while inter-segment points are not. RECURVE works by first deriving a representation trajectory using a time-series representation learning method, then calculating the curvature at each point to identifying boundaries.", "questions": "$\\textbf{Change metric quality: }$ In section 4.3, the authors show that the change metric of RECURVE is on average always high for transition points while distance-based metrics have varying score values depending on the state change. My question is why having varying degrees is a bad thing? Because in many cases, this can also show us the similarity or difference between the two transition states (Walking to sitting vs walking to lying down). \n\n$\\textbf{Scores over time: }$ Please include a figure that shows the estimated CPD scores for a sample over time. This will be very helpful to understand the behaviour of the method, in particular around the change point boundary.\n\n$\\textbf{Error margin: }$ The choice of error margin seems to be an important one, especially to show how the proposed method behaves in slow-transition states. Can you provide performance results across multiple error margin values in the appendix and discuss your observation.", "limitations": "Overall, the limitations are well-covered in the paper. Some points that I mentioned in earlier sections can be added to this discussion.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "$\\textbf{Presentation: }$ The paper is well-written and easy to follow. The graphics are informative and the evaluation has an extensive set of tests.\n\n$\\textbf{Problem definition: }$ The problem this paper is tackling is an important one. Identifying states and state changes in time series is very important for many application. Most change point detection methods are designed to capture abrupt changes in data distribution which is not inline with the reality of state transitions in real-world data. The proposed method claims to capture such gradual transitions better than these existing methods.", "weaknesses": "$\\textbf{Class separated representations: }$ The theoretical analysis relies on Definition 3.4 which states that contrastive-based representations learning produces class-separated representations. First, there is no guarantee of class separation for contrastive-based methods, and in many complex datasets this doesn't in fact happen. If the proposed method is only justified under class-separated settings, this should be clearly stated and the authors should propose a way to assess if the representations generated by a contrastive framework qualify for this. Second, In the case of perfect class separation, a clustering method can be used to identify the underlying states for all windows and therefore find the time of transition between the states. Can the authors justify their reasoning regarding this issue?\n\n$\\textbf{Normalizing the change metric: }$ A limitation of many change point detection (CPD) methods is that they normalize the change metric in order to obtain a normalized score. This means the largest change in a single time series sample will always be assigned a CPD score of 1, even if the time series sample has no true change in state. This also implies that the value of the CPD scores are not comparable across samples because they are normalized per sample. This limitation applies to the proposed method as well and the authors need to elaborate on this issue more. Aside from that, this problem will also have implication on the choice of threshold used to identify the change points. \n\n$\\textbf{Baselines: }$ An important baseline that is missing from the evaluations is a distance-based change point detection model that uses the same representation learning framework as RECURVE. The main claim of the paper is that the curvature is a better measure of change, so this needs to be evaluated in the evaluation section. I would suggest having baselines similar to RECURVE+TNC and RECURVE+TPC, but only using a similarity metric like cosine similarity or even euclidian distance (DISTANCE+TNC and DISTANCE+TPC). Such baseline can also show the advantage of the proposed method in samples with gradual transition.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you so much for acknowledging the impact and intuitiveness of our new boundary detection method.  We hope that our responses have addressed your concerns and that you are able to raise your rating.**\n\n---\n\n**`W1`** *Theoretical analysis with class-separated representations.*\n\n**(1)** You are right. There is no guarantee of class separation, **even though it tends to hold in realistic settings [18, 19]**. As you might agree, theoretical formulation often needs some kind of assumptions on data distribution. Without the class confinement assumption, RECURVE is empirically proven to work well with real-world datasets, because point representations are at least densely populated for intra-segment points and sparsely populated for inter-segment points, under the reasonable quality of the learned representations. Figures 5, 6, and 7 show that the curvature-based scores are higher for inter-segment points than for intra-segment points. In particular, the 2-dimensional plots in Figure 5 show the class-separated representations.\n\n**(2)** In fact, we did try the clustering approach. Our experience indicates that there is no guarantee of one-to-one correspondence between clusters and classes. For example, under the \"Walk\" class, multiple clusters are formed depending on the speed of walking. Thus, clustering (i.e., proper setting of the number of clusters) was not that straightforward for this purpose.\n\n---\n\n**`W2`** *Normalization of the change metric.*\n\nBecause change point detection (CPD) methods are typically applied to **long, continuing time series** (e.g., smartphone sensor recordings for a full day), it is reasonable to think that there are state changes. **The same normalization is applied to *all* samples from an individual dataset by treating them as a single time series**. Thus, we believe that all your concerns on the normalization should be resolved now. The details on the normalization will be clarified in the final draft. Similarly, the threshold $\\varphi$ is set for each individual dataset, as described in Section 3.3.\n\n---\n\n**`W3`** *Missing baselines.*\n\nThank you for your comment. We already have included such baseline for comparison. In fact, what you refer to as DISTANCE(cosine)+TPC is TS-CP$^2$. Per your comment, we add similar baselines, DISTANCE(cosine)+TNC, DISTANCE(Euclidean)+TPC, and DISTANCE(Euclidean)+TNC. Please refer to Table 5 of the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf). Again, **RECURVE is shown to outperform all such baselines**. Specifically, in the HAPT dataset with $p=5$, the increase in the AUC from TS-CP$^2$ to RECURVE+TPC for gradual changes is 32\\%, which is significantly higher than the 21\\% increase for abrupt changes. This finding indicates that RECURVE is notably more efficient in handling gradual changes.\n\n---\n\n**`Q1`** *Quality of the change metric.*\n\nThis is a really good question! As long as the score values of inter-segment points are mostly higher than those of intra-segment points, having varying degrees is not a bad thing, as you thought. However, as shown in the left plot of Figure 1, a wide range of the score values of inter-segment points may overlap with the score values of intra-segment points, especially for gradual changes. Thus, **the accuracy of change point detection is degraded by this overlap in the distance-based method**.\n\n---\n\n**`Q2`** *Figure for the scores over time.*\n\nAbsolutely. Please see Figure 8 of the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf). **The CPD scores of RECURVE indicate the inter-segment points much more clearly than those of TS-CP$^2$**.\n\n---\n\n**`Q3`** *Performance results across multiple error margins.*\n\nTable 2 already contains the performance results across multiple error margins. Per your suggestion, **we present the performance results separately for gradual and abrupt changes across multiple error margins in Table 5** (see the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf)). As the error margin $p$ increases, the AUC metric increases slightly more for gradual changes than for abrupt changes, though the difference is not very clear. We conjecture that the transition strength is a significant factor in distinguishing between gradual and abrupt changes, rather than the transition duration. Table 5 will be added to the final draft.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We deeply appreciate your considerate feedback on our paper. Overall, **we are delighted that most of the reviewers agreed with three main contributions**: **(1) novelty** &mdash; "the idea of leveraging the curvature of a representation trajectory is novel and innovative" (Reviewers EoQb, ZR9k, vKgw, and aPkZ); **(2) effectiveness** &mdash; "RECURVE outperforms state-of-the-art methods" on diverse real-world datasets (Reviewers EoQb, ZR9k, vKgw, and aPkZ); **(3) thoroughness** &mdash; "the feasibility of the proposed algorithm is analyzed intuitively and theoretically" (Reviewers EoQb, ZR9k, and aPkZ). To further clarify our contributions, we have addressed the issues raised as follows.

* **Assuring necessity of the curvature** (Reviewers EoQb, ZR9k, vKgw, and PiYP): We have elaborated the claim that the curvature is essential for handling gradual change points which distance-based metrics cannot capture. Moreover, we support this claim with Table 5 and Figure 9 in the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf).
* **Elucidating theoretical assumptions** (Reviewers EoQb, vKgw, aPkZ, and PiYP): We explain the rationale behind assumptions and justify each assumption using recent literature. 
* **Enriching experiment results** (Reviewers ZR9k, vKgw, and PiYP): To further demonstrate RECURVE's superiority over other change point detection methods, we have added three more baselines and evaluated each method with respect to the category of change points (refer to Table 5 in the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf)).
* **Clarifying visualization** (Reviewers EoQb, vKgw, and PiYP): We eliminate confusions in Figure 5 by describing the configurations for visualizing the trajectories. We also visualize change metric values over time in Figure 8 of the PDF file [🔗](https://openreview.net/attachment?id=XDnlT4Yx3m&name=pdf).

We sincerely hope that our responses have satisfied the reviewers. Also, we welcome further discussions throughout the discussion period. Thank you again for your time and effort!

Meta-review: This paper introduces a novel method, RECURVE, which utilizes the curvature of representation trajectories to detect boundaries in time series data. After a comprehensive evaluation, I recommend accepting this paper for the following reasons. The primary strength of this paper lies in its innovative approach to boundary detection. By analyzing the curvature of representation trajectories rather than relying solely on distance-based metrics, the method effectively handles both gradual and abrupt changes in time series data. This approach addresses a critical gap in existing methods, which often struggle with gradual transitions. The authors provide strong theoretical foundations for their approach, supported by rigorous empirical validation across diverse real-world datasets, where RECURVE consistently outperforms state-of-the-art methods. The paper is also well-structured and clearly written, making it accessible to a broad audience. The authors have responded effectively to reviewer concerns, particularly in demonstrating the necessity of the curvature metric and expanding their experimental comparisons to include additional baselines. They have also provided thorough explanations of the limitations and potential areas for future work, such as exploring class-separated representations further. While some concerns remain about the dependency on the quality of learned representations and the need for clearer guidance on parameter sensitivity, the overall contribution of the paper is significant. It introduces a new perspective on boundary detection in time series, backed by strong empirical evidence and theoretical insights. The method's robustness across different types of changes and datasets suggests that it has the potential to impact various applications significantly.
Decision: Accept (poster)