Optimal Information Retention for Time-Series Explanations
Authors: Jinghang Yue
Jing Wang
Lu Zhang
Shuo Zhang
Da Li
Zhaoyang Ma
Youfang Lin
Venue: ICML 2025 poster (2025)
Keywords: Explanations
Time Series
Optimal Information Retention
Primary area: applications->time_series
TL;DR: We propose the principle of optimal information retention to guide the interpretation of deep models for time series data, and theoretically derive the objective function and propose practical solutions.

Abstract: Explaining deep models for time-series data is crucial for identifying key patterns in sensitive domains, such as healthcare and finance. However, due to the lack of unified optimization criterion, existing explanation methods often suffer from redundancy and incompleteness, where irrelevant patterns are included or key patterns are missed in explanations. To address this challenge, we propose the Optimal Information Retention Principle, where conditional mutual information defines minimizing redundancy and maximizing completeness as optimization objectives. We then derive the corresponding objective function theoretically. As a practical framework, we introduce an explanation framework ORTE, learning a binary mask to eliminate redundant information while mining temporal patterns of explanations. We decouple the discrete mapping process to ensure the stability of gradient propagation, while employing contrastive learning to achieve precise filtering of explanatory patterns through the mask, thereby realizing a trade-off between low redundancy and high completeness. Extensive quantitative and qualitative experiments on synthetic and real-world datasets demonstrate that the proposed principle significantly improves the accuracy and completeness of explanations compared to baseline methods. The code is available at https://github.com/moon2yue/ORTE_public.

Optimal Information Retention for Time-Series Explanations
Jinghang Yue! Jing Wang 12
Lu Zhang'
Shuo Zhang'
Da Li'
Zhaoyang Ma Youfang Lin 13
Abstract
1. Introduction
The rapid advancements in deep learning have greatly im- proved the analysis of time-series data, excelling particu- larly in capturing intricate temporal patterns and managing high-throughpun datasets. However, their inherently multi- layered and nonlinear architecture often renders them black- box, prompting significant concerns regarding interpretabil- ity and reliability in sensitive domains, such as healthcare (Alqaraawi et al., 2020) and finance (Mokhtari et al., 2019). For instance, explaining the temporal patterns and complex dynamics underpinning deep model predictions essential for understanding the physiological representation of dis- eases and enabling effective early warning systems (Di Mar- tino & Delmastro, 2023).
Explaining deep models for time-series data is crucial for identifying key patterns in sensitive domains, such as healthcare and finance. How- ever, due to the lack of a unified optimization criterion, existing explanation methods often suf- fer from redundancy and incompleteness, where irrelevant patterns are included or key patterns are missed in explanations. To address this challenge, we propose the Optimal Information Retention Principle, where conditional mutual information defines minimizing redundancy and maximizing completeness as optimization objectives. We then derive the corresponding objective function the- oretically. As a practical framework, we intro- duce an explanation framework ORTE, learning binary mask eliminate redundant informa- tion while mining temporal patterns of explana- tions. We decouple the discrete mapping process to ensure the stability of gradient propagation. while employing contrastive learning to achieve precise filtering of explanatory patterns through the mask, thereby realizing a trade-off between low redundancy and high completeness. Exten- sive quantitative and qualitative experiments on synthetic and real-world datasets demonstrate that the proposed principle significantly improves the accuracy and completeness of explanations com- pared to baseline methods. The code is available at http://gthtubbco/mmoooyyyuuro public.
Research on improving the interpretability of deep models for time series signals primarily focuses on local explana- tions, aiming to identify critical temporal patterns by lo- cating significant positions in time series signals. These approaches have attracted considerable interest due to their ability to provide intuitive, data-driven insights, highlighting which features are essential or irrelevant for specific down- stream tasks. Some existing local explanation methods are adaptations of approaches originally developed for computer vision (CV) and natural language processing (NLP). How- ever, Ismail al. (2020) illustrates that saliency-based meth- ods, such Integrated Gradients (Bachrens et al., 2010), Deep SHAP (Scott et al., 2017), and Feature Occlusion (Zeiler & Fergus, 2014), fail to reliably and accurately iden- tify the temporal variation in feature importance within time series data. Others are tailored specifically for time-series data. For example, Dynamask (Crabbé Van Der Schaar, 2021) incorporates temporal dependencies by learning the effects of perturbation operators, enabling it to generate instance-specific importance scores for each feature at every time step. TIMEX (Queen et al., 2024) trains surrogate model to replicate the predictive behavior of a black-box model, enabling the identification of important positions and temporal patterns. However, these heuristic methods lack a unified optimization framework and often overlook the requirements for explanation completeness low redun- dancy, posing challenges for identifying critical temporal patterns. As illustrated in Figure 1, the mixing of redundant patterns makes critical patterns unable to be highlighted, while incompleteness may cause missing patterns or even
SShooo of Computer Science and Technology, Beijing Jiaotong University, Beijing, China Key Laboratory of Big Data & Artff- cial Intelligence in Transportation (Beijing Jiaotong University), Ministry of Education, Beijing, China Beijingg Key Laboratory of Traffic Data Mining and Embodied Intelligence, Beijing, China. Correspondence to: Jing Wang <ww@@juu.du...
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Optimal Information Retention for Time-Series Explanations
Effective Information
Redundaut Infornation
#(X*))
1X:X" H(X")
H 1(X
H(X"
HOM
н
H(0
H(X
X
x
X
X
(b)
(a)
Figure 1. The Venn Diagram of information retention. (( ) is defined as task-relevant effective information. (a) Ideal case: X only contains the same information I(X;Y) Y). J. (b) Incompleteness case: The information of X*" a proper subset [(X; Y). (c) Redundant case: The information of X contains not only 7 ((X;)) Y), but also other information irrelevant to Y. (d) Mixed case: A mixture incompleteness and redundant cases.
pointing to wrong patterns.
for heuristic methods in generalizing to time series signals. Therefore, uniform optimization principles and practical frameworks for time series signals needed to be directly applied to improve the local interpretation of deep models.
Information theory provides novel perspective for explain- ing the complex dynamics of deep models. Representative approaches, such as the Information Bottleneck (IB) (Tishby & Zaslavsky, 2015), focus on minimizing the mutual in- formation [(;; Z) between the input X and the learned representation Z, while maximizing the mutual information J(Y; Z) between Z and the labels Y. The IB framework structurally characterizes the learning process as trade-off between compact representations and informational utility. TIMEX+ (Liu et al., 2024) highlights that many existing interpretability metrics for time-series data are prone to triv- ial solutions and issues arising from distribution shifts. To address these challenges, an optimization objective for in- terpretability derived from IB. Contrastive learning, on the other hand, centers on minimizing the distance between positive pairs in the embedding space while maximizing the distance between negative pairs (Dosovitskiy et al., 2014; Khosla et al., 2020; Tian et al., 2020). For instance, train- ing models to maximize the mutual information of positive pairs and minimize that of negative pairs aligns well with this framework (He et al., 2020). Rate reduction focuses on compressing intra-class informational bits, ensuring that representations of different classes occupy highly uncorre- lated linear subspaces (Yu et al., 2020; Chan et al., 2022). CRATE (Yu et al., 2023) interprets the Transformer model as multi-layer stack of information compression and dis- cretization processes, and introduces a simplified, wbite-box version of the Transformer model. While these methods pro- vide insights into the learning process or objectives of deep models from various perspectives, they are not designed to directly guide the optimization of local explanations. Due to the dynamic nature of time series, redundant incomplete local segments may represent different temporal patterns. Additionally, the heterogeneity of data poses challenges
To address the above challenges, we propose the Optimal Information Retention Principle grounded in information theory for precisely identifying explanatory temporal pat- terns within time series. This principle encompasses three criteria to guide the optimization of local explanations: Semantic Information Retention, which aligns the predic- tion distribution of the interpreter with the black-box model to ensure fidelity of explanations; ii) Minimum Redundant Information Retention, which minimizes the mutual infor- mation between the input and the explanation under condi- tions unrelated to downstream tasks, thereby avoiding the inclusion of redundant information; iii Maximum Effective Information Retention, which ensures that after removing the explanatory temporal patterns, the mutual information between the input and the label is minimized. This im- plies that the input loses its predictive capability, thereby guarantecing the completeness of the explanation.
Based on the Optimal Information Retention Principle, we have derived objective functions for distribution alignment, redundancy elimination, and information completeness, re- spectively. Furthermore, we propose Optimal Informa- tion Retention for Time-Series Explanations (ORTE) as a practical framework. Specifically, we first construct a parametric network to learn a binary mask for filtering re- dundant information from the input. We decouple the for- ward mapping and backward gradient propagation processes to mitigate the issue of gradient instability in binary mask learning. Then, utilize the binary mask to construct positive and negative sample pairs and employ contrastive learning to achieve trade-off between low redundancy and
2

---

Optimal Information Retention for Time-Series Explanations
high completeness in the explanations. Additionally, we align the prediction distribution of the explanations with the black-box model to ensure the fidelity of the explanations. Our contributions are summarized as follows:
The probability distribution of M can also be regarded as salient map (attribution map importance map).
2.2. Optimal Information Retention
For a task T whose goal predict label Y from the input X, the optimal sub-feature X X should contain min- imal irrelevant and maximum effective information to the mapping f X + Y. In order to introduce the optimal explanations X", we first investigate the possible cases of information retention within X as shown in Figure 1. Ide- ally, the optimal explanation XTm exclusively and entirely contains the task-relevant information within X, denoted as Figure (a)). Figure (b) shows the incompleteness in- formation case, where XT is a subset of / (X;; (X; failing to fully explain the predictive behavior of fX)). This may manifest in the explanation as only highlighting partial frag- ments of the effective series. On the other hand, a redundant information case occurs when Xm not only includes the necessary explanatory information but also incorporates ir- relevant information, as shown in Figure 1(c). This may lead to the mixing of irrelevant patterns, causing critical tem- poral patterns to fail to be highlighted. Figure 1 (d) involves combination of incompleteness and redundant information within a single context, which may be a more challenging case. Both the inclusion of redundant information and the absence of effective information can alter critical temporal patterns, which is a key challenge in time-series explana- tions. Furthermore, explanations should remain faithful to the pre-trained black-box model. To this end, we propose the optimal information retention principle containing three criteria: Semantic Information Retention, Minimum Re- dundant Information Retention, and Maximum Effective Information Retention, as outlined below.
We propose the Optimal Information Retention Prin- ciple grounded in information theory to address the issues of redundancy and incompleteness in time series interpretation, and theoretically derive the correspond- ing objective functions.
We propose ORTE as practical framework, which achieves trade-off between low redundancy and high completeness in explanations by decoupling discrete mapping, contrastive learning, and distribution align- ment.
We achieve state-of-the-art performance on eight syn- thetic and real-world time series datasets compared to the latest competitive baselines and validate the utility of each component.
2. Optimal Information Retention Principle
In this section, we first formally describe the formulation of time-series explanations based on classification task. Then, we present a unified optimization principle for time-series explanations, namely the principle of optimal information retention. This principle comprises three criteria: semantic information retention, minimal redundant information reten- tion, and maximal effective information retention. Based on these criteria, we theoretically derive objective functions for optimizing time-series explanations.
2.1. Problem Formulation
Criteria 2.1. (Semantic Information Retention) To faith-
Time Series Classification, Given time series dataset D = (X, Y) = {(1 b containing N pairs, X are multivariate or univariate time series instances, and Y are the corresponding labels. Each time series instance T; E T is composed of vectors collected by one (e.g., D 1) or multiple sensors (e.g.. D > 1), where T denotes the length of time steps and D represents the number of sensors. A pre- trained deep neural network (DNN) f is trained to classify the input instance 2; into the maximum probability class, e.g., Di- = argmaz(f f (x:)) E C.
fully reflect the predictive behavior f(X), X should per- form similar task-specific semantic consistency, as follows:
min Las (J(X), (X")),
(I)
where LJs denotes the Jensen-Shannon divergence.
Each label y; belongs to one class of (1,2....,C') C).
Criteria 2.1 ensures the fidelity of the explanation by align- ing the distributions of f(X) and f(X"). This avoids spuri- ous correlations between the interpreter and the pre-trained model, meaning that the identified key temporal patterns are indeed the basis for the black-box model's ''s predictions. Furthermore, Semantic Information Retention also aligns with the advantages of prediction consistency emphasized in TIMEX (Queen et al., 2024) and the signal issue addressed in TIMEX+ (Liu et al., 2024).
Local explanation for time series. In this paper, we aim to propose mask generator 9(f,z) to learn a binary mask matrix M E T) which can retain the important pieces of I, e.g., Xm = X @ M, where @ is an element- wise multiplication. In other words, X are considered more important than X 3 (1 - -M) for the classification task. Naturally, the ideal sub-features X are expected to retain the most relevant information to the downstream task.
Criteria 2.2. (Minimum Redundant Information Retention)
The optimal sub-features X should only contain the mu- tual information shared by T(X;Y), and include as little
3

---

Optimal Information Retention for Time-Series Explanations
information of X no relation Y, as follows:
To minimize I TX; Y xx),, we minimize the first term I (x:Y, while maximize the second term ((;;X)
min I X" Y).
For the first term, when X serves as the input generated by XTm, the mutual information [(X; X) should ideally include only the information contained in [X; Y). Other- wise, redundant information may be introduced. Assuming X has no overlap with the portions of H(X | Y), then
Although Criteria 2.2 provides an ideal direction for mini- mizing redundant information, computing the conditional mutual information remains difficult for:
p (x:"
X"; Y)= Ep(x.y)
(p((e=la.0)) log
I (x:X, = 1 (X;Y), which is a constant independent of X.
p (2 y)
(3)
where p (x" x, 3) and p (x=" y) are intractable in DNN. Considering Xm = X O M, I (X; M | YY) is served as the simplified form of / (X; X Y). Furthermore, by relating it to the Kullback-Leibler divergence, there are
For the second term, since X and X share effective informa- tion in X", they can naturally regarded as the anchor sam- ples and the positive samples, respectively, in contrastive learning. Therefore, we employ contrastive loss to maxi- mize the mutual information between X and X:
I [X; M Y) = Ep(a,s) (pkm x, lp(m y))l,
(4)
where I XX; IY) x (pm( x, v)lp(( v))- Referring to the variational approaches (Miao et al., 2022; Queen et al., 2024), M can be sampled following Bernoulli distribution at each time step, ie.,, p(m x) = a Bern(pt.a) where Pt.d denotes the probability that the value of time and dimension d is retained. Assuming the desired distribution of p is p(m) III,a Bern(r), the derived objective function can be expressed as:
1 N N
Lat
Σ (loglexp (f (zx) o (f.))
K
loglexp (f (x:) (i:)) + Σ exp (f (z,) ff (1))
(8)
where $j denotes the negative sample, which can be gener- ated by 1((X - X XM), i.e., Gaussian noise imputation. The detailed proof of (8) is provided in Appendix B.2.
Lmaas = E[DRI. (p(m [x)]((m))]
> 1.d
Ppr.a log
Pt.d + (1 Pt,a) log
1- P. (5)
r
1 1-r r
3. ORTE Method
We now present the Optimal Information Retention for Time-Series Explanations (ORTE), whose overview frame- work is illustrated in Figure 2. Based on the analysis in the previous section, ORTE describes a practical framework for time-series explanations from the optimal information retention principle. ORTE includes three main blocks like Adaptive Mask Generator (AMG), Contrastive Explanation Separator (CES), and Predictive Distribution Aligner (PDA), Specifically, AMG learns a 0-1 occlusion matrix to mask redundant information so that achieves the Minimum Re- dundant Information Retention. CES employs contrastive learning to trade off low redundancy and high complete- ness in the explanations, thereby achieving the Maximum Effective Information Retention. PDA aligns the prediction distribution of the explanations with the black-box model to ensure the Semantic Information Retention.
where r as the desired probability of q(m = is is hyper- parameter in practice. The detailed proof of (5) is provided in Appendix B.I.
Intuitively, the optimal X should contain all the in- formation relevant to Y. Assuming Xm is removed, (X;Y| XX)) + O indicates that H(X xT") completely loses the information required to predict Y. To this end, the formal definition of maximum effective information retention is as follows.
Criteria 2.3. (Maximum Effective Information Retention)
The optimal retention of sub-features X should encom- pass as much telk-relevant information from X as possible, minimizing the risk of information omission, as follows:
min (X;Y X
We pad the masked parts of X to obtain X, ensur- ing consistency in dimensions while avoiding the Out-of- Distribution (OOD). Furthermore, (6) can be decomposed as:
3.1. Adaptive Mask Generator
According to the Minimun Redundant Information Reten- tion, the binary mask M E {0, reveals the inter- pretability of the time series, i.c., important segments are assigned {1} to be retained. The explanation generator is performed through two steps to learn M: Probabilistic Generation Process and Discrete Mapping Process.
1 Y Ixm)
7 7xx: 1 =1 1x -xxxxx,)= (15.x)-((xxx0)) -xxx))

---

Optimal Information Retention for Time-Series Explanations
Contrastive Explanation Separator
Predictive Distribution Aligner
Adaptive Mask Generator
Inputs
Explanations
Inputs
Logits
Inputs
Inputs
4
Black-box
MI P
of of Transformer
Positive Samples
Class
Negative Samples
soffman
Discrete
Noisz Padding
Class 1. Class 2.....
Class
Mapping
Black-box
M
1 M
Cuprtt+ Ylepe
Inputs
Positive Sumples
Figure 2. Overview of ORTE method including three main blocks: Adaptive Mask Generator (AMG), Contrastive Explanation Separator (CES), and Predictive Distribution Aligner (PDA).
Probabilistic Generation Process. The first process en- codes the input X as a probabilistic matrix P(M X), where Ptr.d, > Ptz,da indicates It,.d; is more important than d2 As discussed by Queen et al. (2024), em- ploy an autoregressive transformer decoder and softmax activations to output the probability of each temporal sen- sor pair. Additionally, to optimize the continuity of the distribution, we also introduce a continuous loss:
pass is:
OC OC 0m Op OC 0p 8m 0t
backward:
2
(11)
Ol 3m Op al
Estimator (STE) (Bengio et al., 2013) is applied to allow gradients to propagate through the logits, such as Sp Rs Although ST-GS solves the prob- lem of non-diffeerentiable gradients in the discrete mapping process, it ignores the difference between forward and back- ward propagation. Specifically, the forward propagation requires the distribution of to be sharp with the training process so that extreme values are taken without signifi- cant bias. However, as discussed in Shah et al. (2024) the backpropagation process of the fixed gradient limits the ability to model the data distribution. We propose adaptive STE (adapt-STEB to alleviate the asymmetry of forward and backward propagation:
where
D T-1
Leon TXD ΣΣ
V (pr.d PP+1.)
So the objective function for minimal redundant informa- tion retention can be expressed fmair Lmask Lcon- Discrete Mapping Process. Mapping P from the contin- uous space to the discrete binary space M suffers from non-differentiable gradients. Queen et al. (2024) introduces Straight- Through Gumbee-SSfttax (ST-GS) estimator (Jang al., 2016) bridge the gap between the forward propagation mapping discrete distribution and the backward derivation being bon-differentiable Let I denotes the unnor- malized logits of encoder outputs, the forward pass can be summarized as:
Straight-Through
7 = clip (T/ - 17 (l),, Tmins Tmax)
7 clip (Ty 1) a(I), Twin Tmax); (12)
where Ty and T. represent the temperature parameter of the forward propagation and backward propagation, respec- tively. clip() restricts the upper and lower bound hyper- parameters of T and TO be Tmax and Tmins and is the updating rate and (7) () is the standard deviation of l. A small Ty ensures the sharpness of in the forward propagation, while a large % ensures the faithfulness to the backward propagation. The pseudocode of adapt- is summarized in Appendix C
(0)
forward: softmax
(10)
m arg mp)
where T represents the temperature parameter that controls the entropy of p, and is randomly sampled noise from Gumbel 10. Giving the loss function, the backward
5

---

Optimal Information Retention for Time-Series Explanations
3.2. Contrastive Explanation Separator
datasets and four real-world datasets. Without loss of gener- ality, we employ the vanilla Transformer (Vaswani, 2017) as black-box classifier. Our goal is to explain this black-box model. Here we first introduce the experimental settings including Datasets, Metrics, and Comparison Methods. Then the performance of the proposed method is compared with baseline methods and an ablation study is conducted to validate the effectiveness of individual modules. Finally, the ability of the optimal information retention principle distinguish valid information from redundant information is experimentally verified. Refer to the Appendix D for more experimental details.
As discussed in Section 2.3, maximizing the task-relevant information shared by positive samples X with input X, and minimizing the retention of valid information by negative samples X is key to interpretation. To this end, we intro- duce how to generate positive and negative sample pairs. As discussed in TIMEX+ (Liu et al., 2024). it is essential to ensure the integrity of effective information while avoiding out-of ditt----ittribuiionn (OOD) issues. Therefore, we adopt the same generation strategy:
X = M),(M,X),,
(13)
where X E RIXDD Liim mse(X, X) is employed to ensure generalization and avoid overfitting. For negative samples X- we filter out effective information through reverse masking, while the masked portions are then filled with Gaussian noise:
4.1. Experimental Settings
Datasets. We evaluate our method on four carefully de- signed synthetic datasets with ground-truth annotations: FreqShapes, SeqComb-UV, SeqComb-MV, and Low Var (Queen et al., 2024). These datasets, encompassing both univariate and multivariate time series, inherently contain complex temporal dynamics. The goal of the explanation is to identify key segments that represent critical patterns in the data. Additionally, we test on four real-world datasets: ECG - ECG arrhythmia detection (Moody & Mark, 2001);
= (1) pad,
(14)
where pad- ~ N' (4, This effectively enhances the discrepancy between positive and negative samples, thereby facilitating a clearer distinction in the feature space. So the objective function for maximum effective information retention can be expressed as Lmeir = aeLet + BLeii where and 3 are hyperparameters.
PAMM - human activity recognition (Reiss & Stricker, 2012);
Epilepsy EEG seizure detection (Andrzejak et al., 2001);
and Boiler mechanical fault detection (Shohet al., 2019).
For the ECG dataset, we define the ground-truth explanation as the QRS intervals associated with arrhythmia detection. Makowski et al. (2021) provides standard reference points for the detection of R, P. and T waves in ECG signals.
3.3. Objective Function
Predictive Distribution Aligner. The explanation should be faithful to the predictions of the black-box model. For this purpose, we align the explained predictive distribution with the pre-trained black-box model predictive distribution. Referring to 1, the Jensen-Shannov divergence is minimized as the loss function to reduce the distribution difference.
Metrics. We employ two evaluation approaches. For expla- nations with ground truth annotations, we use the labeled explanations as a reference and quantify the results using Area Under Precision (AUP), Area Under Recall (AUR), and Area Under Precision Recall Curve (AUPRC), which combines AUP and AUR. For all three metrics, higher val- ues indicate better performance. For explanations without ground truth annotations, we follow an approach similar to Queen et al. (2024), where the bottom p-percentile of fea- tures identified by the explainer is masked, and the change in predictive Area Under Receiver Operating Characteristic Curve (AUROC) is measured.
The overall objective function for the learning process con- sists of four terms. First, LJs aligns the explanation with the predictions of the black-box model, ensuring the retention of semantic information. Second, Lmire filters redundant information by learning a masking matrix. Third, Lmeir de- composes the original data into positive and negative sample pairs, where contrastive learning is used to preserve effec- tive information. Additionally, to enforce sparsity in the masking, introduce Exp || as regularization term. The overall objective function is as follows:
Comparison Methods. We have compared the proposed method with seven explainability methods, as follows: In- tegrated gradients (IG) (Baehrens et al., 2010) is widely used for general interpretation; Dynamask (Crabbé & Van Der Schaar, 2021) and WinIT (Leung et al., 2021) are the time-series specific explanation methods; CoRTX (Chuang et al., 2023) is explanation method based on contrastive learning: SGT + Grad (Ismail et al., 2021) is an in-hoc inter- preter for time-series data; TIMEX (Queen et al., 2024) em- ploys a white-box model as an interpreter; and TIMEX+
C = Los + Lmrit + Cmeir + Caper
(15)
where y is hyperparameter.
4. Experiments
In this section, we conduct experiments to study the per- formance of our proposed ORTE method on four synthetic
6

---

Optimal Information Retention for Time-Series Explanations
Table Attribution explanation performance on univariate and multivariate synthetic datasets.
FREQSHAPES
SEQCOMB-UV
METHOD
AUPRC
AUP
AUR
AUPRC
AUP
AUR
IG
0.7516+000032
0.6022+0.008
0.5975+0000020
0.5760+0.0002
0.8157+0.0023
0,2868
DYNAMASK
0.220110.0011 0.2952=0.003
0.5037=000015
0.8782:10.0033 0.4421
0.1029=0.0007
WINIT
0.5071 0.5546=0.0022
0.4557=0.0016
0.4568=0.0017
0.7872++0.022
0.2253
CORTX
0.00161000156 0.493810.0000
0.326110.0012
0.5643110.0024
0.824110.0025
0.1749
SGT + GRAD
0.5012:10.019 0.413810.0011
0.3931=0.0015
0.0031=0.0021
0.7028=0.0013
0.213611.0008
TIMEX
0.8324+0.0088 0.7219+0.0033
0..001-000022
0.7124+000017
0.9411+00.000
0.3380++0.0014
TIMEX+
0.8905++0.00
0.7805+0.000
0.6618-00.019
0.8468 4..0044
0.9069-40.000
0,4064-10.001
ORTE
0.999
0.8269
0.8298=000020
0.9001=0.0025
0.9711#0.0006
0,4503+0.0031
SEOCOMB-MV
LOWVAR
METHOD
AUPRC
AUP
AUR
AUPRC
AUP
AUR
IG
0.329810.00155
0.7483=0..022
0,2581 0028
0,8691100.0335
0.4827+0.0029
0.816510.0016
DYNAMASK
0.003611.0009
0.5481=0.0055
0.1953
0.1391#0.0012
0.1640+000028
0.2106+0.0018
WINIT
0.2809 10.0018
0.759410.0022
0.007700.022
0.166710.0015
0.1140:10.0022
0.3842
CORTX
0.3629 1020
0.562510.000 6,0006
0.345710.000
0,498311000014
0.0281
0.4011
SGT GRAD
0.4893+000000
0.4970+0.00M
0.4289+00001
0.3449++00010
0.2133+000029
0.0008
TIMEX
0.6878+00.022
0.8326+0.0000
0.3872+-0.0015
0.8673+000033
0.5451+000008
0.9004+00.024
TIMEX++
0.7589 0.8783#0.0007
0.3906=000001
0.9466=00.0015
0.8057
0.8332+00.006
ORTE
0.831410.0019
0.9011#000005
0.5632:10.002
0.9637=000016
0.7390+00.003
0.9057#0.0013
Table 2. (Left) Attribution explanation performance on the ECG dataset. (Right) Results of ablation analysis.
ECG
ORTE
ECG
METHOD
AUPRC
AUP
AUR
ABLATIONS
AUPRC
AUP
AUR
IG
0.4182+0.0014
0.5949- 0.0023
0.3204++000012
FULII
0.7183+0.000
0.8222+00.002
0.5133++0.0016
DYNAMASK
0.3280++0.0011
0.5249+0.000
0.00000-000000
replace STR
0.6152+ 0.0007
0.7468+0.. 0.00088
0.4023++0.0012
WINIT
0.0001+000011
0.4431+000022
0.3474-+000011
w/o LJs
0.6946=0.0040
0.7941+000005
0.5208+0000036
CORTX
0.3735
0.4968
0.0024
0.3031+0.0000
w/o Lea
0.5421+000003
0.7195 0.0028 0.0002
0.4143++00019
SGT + GRAD
0,3144++0.0110
0.4241 0.0002
0.2639-+0.0013
w/o Crom
0.7130=0.0087
0.8019+0..000 0.0037
0.4244++0.002
TIMEX
0.4021=+0.0018
0.5663=+0.005
0.4457=+0.0018
w/o Excm
0.7017=000000
0.79511=0.0044
0.5187-+0.0035
TIMEX+
0.6099
0.0000=000010
0.4595++0.0007
w/o Lage
0.6977=0.0051
0.3507+000007
0.8185:10.0044
ORTE
0.7089:10.0019
0.5133:10.0166 0.0021
(Liu et al., 2024) is an improved version of TIMEX based on the information bottleneck principle.
achieves an optimal trade-off between low redundancy and high completeness. When considering the global metric AUPRC, ORTE significantly improves ground-trute expla- nations over TIMEX++ with improvements of 12.27% on FreqShapes, 6.29% on SeqComb-UV, 9.55% on SeqComb- MV, and 1.81 % on Low This aligns with our empha- sized principle of optimal information retention. We also provide experimental results on CNN and LSTM in Ap- pendix E and visualizations in Appendix H.
4.2. Performance on Synthetic and Real-world Datasets
Synthetic datasets. As shown in Table 1, we compare ORTE with existing competitive interpreters on both uni- variate and multivariate datasets. ORTE achieves the best performance in 10/12 cases (3 metrics across datasets). Compared to the strongest baseline TIMEX++, ORTE im- proves AUPRC by average of 7.48%, AUP by 1.84%, and AUR by 22.27%. Compared to the second strongest baseline TIMEX, ORTE improves AUPRC by an average of 19.61%, AUP by 15.38% and AUR by 22.33%. Previous methods prioritized the precision of temporal pattern local- ization but overlooked the completeness of explanations, resulting in higher AUP but lower AUR, In contrast, ORTE
Occlusion experiments on real-world datasets. To evalu- ate the explanations of ORTE on real-world datasets lacking ground-truth, we follow the approach of TIMEX (Queen et al., 2024) and TIMEX++ (Liu et al., 2024) by masking the bottom p-percentile of the salient features and measuring the change in prediction AUROC. As shown in Figure 3, ORTE outperforms other methods on the PAM and Boiler
7

---

Optimal Information Retention for Time-Series Explanations
Random
Dynamask
PAM
Epilepsy
Boiler
0.95
Prciclon Ro
Prodiction AUROC
0.00
0.6
0.75
0.85
0.70
0.4
0.80
0.75
0.8
0.85
0.9
0.925
0.95
0.975
0.99
0.75
0.0
0.85
0.925
0.95
0.975
0.99
0.75
0.05
0.9
0.925
0.95
0.975
0.99
Bottom Proportion Perturbed
Bottom Proportion Perturbed
Bottom Proportion Perturbed
Figure
Occlusion experiments real-world datasets. Higher values indicate better performance.
Random
Dynamask
Timex
Timex++
Ours
Epilepsy
1.00
0.95
0.90
Prddiction KUROC
0.90
0.85
0.8
0.85
0.75
0.80
Prediction
0.75
0.65
0.5
0.70
0.60
0.55
0.50
02
0.40
0.75
0.8
0.925
0.975 0.99
0.8
0.85
0.925
0.95
0.975
0.99
0.75
0.05
0.925
0.95
0.975 0.99
Bottom Proportion Retainad
Proportion Retained
Figure 4. Insertion experiments on real-world datasets. Higher values indicate better performance.
4. The results show that ORTE achieves a lower AUPROC when inserting the bottom 75% of the salient features. As the insertion percentage increases, the predicted AUROC gradually improves. When the insertion ratio reaches 97.5%, ORTE attains the highest AUPROC. This indicates that the most interpretable informative features concentrated in the high-saliency features, further validating the claims of low redundancy and high completeness. In addition, the Random baseline always maintains a high AUROC, which is because the random selection of features does not dis- tinguish the salient or importance of features and contains informative time points or segments.
0.6
0.0
FREOSHAPE
SEOCOMB-UV
SBOCOMB-MY
LOWVAR
Figure Comparison performance of AUP between adopt-STE and STE.
4.3. Ablation and Contrast Experiments
Results and Ablation Experiments on ECG Data. To di- rectly demonstrate ORTE's performance on real-world data, compared it with baseline methods in the application of arrhythmia detection using electrocardiogram (ECG) data. As shown in Table 2, the attribution maps of ORTE exhibit state-of of-the-ant performance in identifying the relevant QRS intervals driving arrhythmia diagnosis, outperforming the strongest baseline TIMEX+ by 3.85% (AUPRC), 13.25% (AUP), and 11.71% 1% (AUR). Notably, ORTE achieves an overall improvement in both AUP and AUR, ensuring the identification of larger segments of the QRS interval rather than individual time steps.
datasets, and performs comparably to the strongest base- line TIMEX++ on the Epilepsy dataset, both maintaining a high and stable level. Notably, ORTE demonstrates ex- ceptional stability on the Boiler dataset, retaining high AUROC even after masking 97.5% the bottom features. Furthermore, compared to other baseline methods, ORTE ex- hibits narrower error bars, indicating that the ORTE method, guided by the principle of optimal information retention, also achieves outstanding stability.
Insertion experiments on real-world datasets. As com- plement to the occlusion experiments, the insertion experi- ment inserts features from bottom to top, as shown in Figure
8

---

Optimal Information Retention for Time-Series Explanations
To validate the contribution of each component in the ORTE method, we conducted ablation experiments, shown the right side of Table 2, where "w/o" indicates the ab- sence of the relevant component. The results demonstrate that adapt-STE improves AUPRC by 7.27% and AUR by 14.22%, proving the effectiveness of adapt-STE. We also compared the performance of every term of the objective function. Compared the full model, the absence of other loss functions leads to inferior explanations. The proposed ORTE effectively compensates for the limitations of individ- ual components, achieving an overall optimal explanation.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
References
Alqaraawi, A., Schuessler, M., WeiB, P., Costanza, E.,, and Berthouze, N. Evaluating saliency map explanations for convolutional neural networks: user study. In Proceed- ings of the 25th international conference on intelligent user interfaces, pp. 275-285, 2020.
Contrast Experiments of adapt-STE. We compare the performance of adapt-STE with STE on univariate and mul- tivariate datasets, as shown in Figure The AUP values of the four datasets show that adapt-STE is significantly better than STE. Moreover, AUP is considered to be the more im- portant evaluation metric. Especially in SeqComb-UV and SeqComb-MV adapt-STE can detect the key time patterns more accurately, which also verifies the better adaptation of adapt-STE to data diversity. The complete experiment results are provided in Appendix G.
Andrzejak, R. G., Lehnertz, K., Mormann, F., Rieke, C., David, P., and Elger, C. E. Indications of nonlinear deter- ministic and iiite-dimensional structures in time series of brain electrical activity: Dependence on recording re- gion and brain state. Physical Review E, 64(6):061907, 2001.
Baehrens, D., Schroeter, T., Harmeling, S., Kawanabe, M., Hansen, K., and Müller, K.-R. How explain individual classification decisions. The Journal of Machine Learning Research, 11:1003-188 1831, 2010.
5. Conclusion
This paper proposes an cntormmtiionthhorreee principle of optimal information retention to guide the discovery of ex- planatory temporal patterns in time series. It defines se- mantic information, redundant information, and effective information, and derives corresponding objective functions. Furthermore, we introduce ORTE a practical implemen- tation of this principle. Experiments on synthetic and real- world datasets demonstrate that the proposed method signif- icantly outperforms competitive baseline approaches. We also believe that the optimal information retention principle proposed in this paper is applicable to the optimization and improvement of future interpretability methods.
Bengio, Y., Léonard, N., and Courville, A. Estimating or propagating gradients through stochastic neurons for con- ditional computation. arXiv preprint 2013.
Bento, J., Saleiro, P., Cruz, A. F., Figueiredo, M. A., and Bizarro, P. Timeshap: Explaining recurrent models through sequence perturbations. In Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining, pp. 2565-2573, 2021.
Bhalla, U., Srinivas, S., and Lakkaraju, H. Discriminative feature attributions: bridging post hoc explainability and inherent interpretability. Advances in neural information processing systems, 66:44105-44122 2023.
Limitations. This study has two main limitations. First, the affect instance-level expla-
control of hyperparameters nations, requiring trial and adjustment based on the type and characteristics of the data. Second, the practical application of the optimal information retention principle to guide other interpretability methods requires further research. Extend- ing it to gradient-basee based or mask-based explanation methods could be an interesting direction.
Bhatt, U., Xiang, A., Sharma, S., Weller, A., Taly, A., Jia, Y., Ghosh, J., Puri, R., Moura, J. M., and Eckersley, P. Explainable machine learning in deployment. In Proceed- ings of the 2020 conference on fairness, accountability. and transparency, pp. 448-657,45 2020.
Chan, H. R., Yu, Y., You, C., Qi. H., Wright, J., and Ma, Y. Redunet: A white-bo -oxte--bx deep network from the principle of maximizing rate reduction. Journal of machine learning research, 23(1 23(1 14):111401.3 103, 2022.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 62372031).
Chen, J., Song, L., Wainwright, M., and Jordan, M. Learn- ing to explain: An information-theorrtii perspective on model interpretation. In International conference on ma- chine learning, pp. 883-892, PMLR, 2018
9

---

Optimal Information Retention for Time-Series Explanations
Chen, X., Qiu, P., Zhu, W., Li, H., Wang, H., Sotiras, A., Wang, Y., and Razi, A. Timemil: advancing multivariate time series classification via time-aware multiple in- stance learning. In Proceedings of the 41 International Conference on Machine Learning, pp. 7190-7206, 2024.
Guidotti, R., Monreale, A., Spinnato, F., Pedreschi, D., and Giannotti, F. Explaining any time series classifier. In 2020 IEEE second international conference cognitive machine intelligence (CogMI), pp. 167-176. IEEE, 2020.
He, K., Fan, H., Wu, Y., Xie, S., and Girshick, R. Mo- mentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 9729-9738, 2020.
Choi, E., Bahadori, M. T., Sun, J., Kulas, J., Schuetz, A., and Stewart, W. Retain: An interpretable predictive model for healthcare using reverse time attention mechanism. Advances in neural information processing systems, 29, 2016.
Hooker, S., Erhan, D., Kindermans, P.J., and Kim, B. benchmark for interpretability methods in deep neural networks. Advances in neural information processing systems, 32, 2019.
Chuang. Y.-N., Wang, G., Yang. F., Zhou, Q., Tripathi, P., Cai, X., and Hu, X. Cortx: Contrastive framework for real-time explanation. arXiv preprint arXiv:2303.02794, 2023.
Ismail, A. A.. Gunady, M., Corrada Bravo, H., and Feizi, S. Benchmarking deep learning interpretability in time series predictions. Advances in neural information pro- cessing systems, 33:6441-6452, 2020.
Crabbé, J. and Van Der Schaar, M. Explaining time series predictions with dynamic masks. In International Con- ference on Machine Learning, pp. 2166-2177. PMLR, 2021.
Ismail, A. A., Corrada Bravo, H., and Feizi, S. Improving deep learning interpretability by saliency guided training. Advances in Neural Information Processing Systems, 34: 26726-26732 2021.
Dabkowski, P. and Gal, Y. Real time image saliency for black box classifiers. Advances in neural information processing systems, 30, 2017.
Jang, E., Gu, S., and Poole, B. Categorical repa- rameterization with gumbel-softmax, arXiv preprint 4rXiv:1611.01144 2016.
Di Martino, F. and Delmastro, F. Explainable ai for clinical and remote health applications: survey on tabular and time series data. Artificial Intelligence Review, 56(6): 5261-5315, 2023.
Jethani, N., Sudarshan, M., Aphinyanaphongs, Y., and Ran- ganath, R. Have we learned to explain?: How inter- pretability methods can learn to encode predictions in their interpretations. In International Conference on Arti- ficial Intelligence and Statistics, pp. 1459-1467. PMLR, 2021.
Doddaiah, R., Parvatharaju, P., Rundensteiner, E., and Hartvigsen, T. Class-specific explainability for deep time series classifiers. In 2022 IEEE International conference on data mining (ICDM), pp. 101-110. IEEE, 2022.
Dosovitskiy, A., Springenberg, J. T., Riedmiller, M., and Brox, T. Diseriminative unsupervised feature learning with convolutional neural networks. Advances in neural information processing systems, 27, 2014.
Khosla, P., Teterwak, P., Wang, C.. Sama, A., Tian, Y., Isola, P., Maschinot, A., Liu, C., and Krishnan, D. Supervised contrastive learning. Advances in neural information processing systems, 13:11-66618773, 2020.
Early, J., Cheung, G., Cutajar, K., Xie, H., Kandola, J., and Twomey, N. Inherently interpretable time series classi- fication via multiple instance learning. In The Twelfth International Conference Learning Representations, 2024.
Langley, Crafting papers on machine learning. In Langley, P. (ed.), Proceedings of the 17th International Conference Machine Learning (ICML 2000), pp. 1207-1216, Stan- ford, CA, 2000. Morgan Kaufmann.
Leung, K. K., Rooke, C., Smith, J., Zuberi, S., and Volkovs, M. Temporal dependencies in feature importance for time series predictions. arXiv preprint arXiv:2107.14311 2021.
Fong, R., Patrick, M., and Vedaldi, A. Understanding deep networks via extremal perturbations and smooth masks. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 2950-2958, 2019.
Liu, Z., Wang, T., Shi, J., Zheng, X., Chen, Z., Song, L., Dong, W., Obeysekera, J., Shirani, F., and Luo, D. Timex++: Learning time-series explanations with infor- mation bottleneck. arXiv preprint arXiv:2405.099 09300, 2024.
Guidotti, R., Monreale, A., Giannotti, F., Pedreschi, D., Ruggieri, S., and Turini, F. Factual and counterfactual explanations for black box decision making. IEEE Intel- ligent Systems, 34(6):14 14-3)34-23,,11 2019.
10

---

Optimal Information Retention for Time-Series Explanations
Lundberg, S.
A unified approach to interpreting model predictions. arXiv preprint arxiv::705.00770 1705.07874, 2017.
Selvaraju, R. R., Cogswell, M., Das, A., Vedantam, R.,
Parikh, D., and Batra, D. Grad-cam: Visual explana- tions from deep networks via gradient-based localization. In Proceedings of the IEEE international conference on computer vision, pp. 518-626, 2017.
Makowski, D., Pham, T., Lau, Z. J., Brammer, J. C., Lespinasse, F., Pham, H., Schölzel, C., and Chen, S. A. Neurokit2: A python toolbox for neurophysiological sig- nal processing. Behavior research methods, pp. 1-8, 2021.
Shah, R., Yan, M., Mozer, M. C., and Liu, D. Improv- ing discrete optimisation via decoupled straight-throug gumbel-softmax, arXiv preprint arXiv:2410. 24Vi:24100.1331,, 2024.
Miao, S., Liu, M., and Li, P. Interpretable and generaliz- able graph learning via stochastic attention mechanism. In International Conference on Machine Learning, pp. 15524 15543. PMLR, 2022.
Shohet, R., Kandil, M., and McArthur, lated boiler data for fault detection and classification, 2019. URL hitps:/ /tt/:///xx.o/ doi wrg.l0.gg//00 212277 a awav-bn36.
Simu-
Mokhtari, K. E., Higdon, P., and Başar, A. Interpreting financial time series with shap values. In Proceedings of the 29th annual international conference computer science and software engineering, pp. 166-172,200
Tian, Y., Sun, C., Poole, B., Krishnan, D., Schmid, C.,, and Isola, P. What makes for good views for contrastive learning? Advances in neural information processing systems, 33:6827-6839 2020.
Moody, G. B. and Mark, R. G. The impact of the mit-bih arrhythmia database. IEEE engineering in medicine and biology magazine, 20(3):45-555 50, 2001.
Tishby, N. and Zaslavsky, N. Deep learning and the infor- mation bottleneck principle. 2015 ieee information theory workshop (itw), pp. 1-5. IEEE, 2015.
Mujkanovic, F., Doskoč, V., Schirneck, M., Schäfer, P., and Friedrich, T. timexplain- framework for explaining the predictions of time series classifiers. arXiv preprint arXiv:2007.07606 2020.
Tonekaboni. S., Joshi, S., Campbell, K., Duvenaud, D. K., and Goldenberg, A. What went wrong and when? instance- feature importance for time-series black- box models. Advances in Neural Information Processing Systems, 33:799-809, 2020.
Qin, Y., Song, D., Chen, H., Cheng, W., Jiang, G., and Cottrell, G. A dual-stage attention-basee recurrent neu- ral network for time series prediction. arXiv preprint arXiv: 02971, 2017.
Vaswani, A. Attention all you need. Advances Neural Information Processing Systems, 2017.
Queen, O., Hartvigsen, T., Koker, T., He, H., Tsiligkaridis, T., and Zitnik, M. Encoding time-series explanations through self- supf-supe model behavior consistency. Ad- vances in Neural Information Processing Systems, 36, 2024.
Yosinski, J., Clune, J., Nguyen, A., Fuchs, T., and Lipson, H. Understanding neural networks through deep visual- ization. arXiv preprint arXiv: 00di:/555606655 506.06579, 2015.
Yu, Y., Chan, K. H. R., You, C., Song, C., and Ma, Y. Learning diverse and discriminative representations via the principle of maximal coding rate reduction. Advances neural information processing systems, 33:9422 24444,, 2020.
Räuker, T., Ho, A., Casper, S., and Hadfield-Menell, D. To- ward transparent ai: A survey on interpreting the inner structures deep neural networks. In 2023 ieee confer- ence on secure and trustworthy machine learning (satml),
pp. 464 483. IEEE, 2023.
Yu, Y., Buchanan, S., Pai, D., Chu, T., Wu, Z., Tong, S., Haeffele, B., and Ma, Y. White-box transformers via sparse rate reduction. Advances Neural Information Processing Systems, 36:9422-9457, 2023.
Reiss, and Stricker, D. Introducing a new benchmarked dataset for activity monitoring. In 2012 16th international symposium wearable computers, pp. 108 109. IEEE, 2012.
Zeiler, M. and Fergus, R. Visualizing and understanding convolutional networks. d. fleet, t. pajdla, b. schiele, & t. tuytelaars (eds.). computer vision-eccv 2014, 2014.
Ribeiro, M. T., Singh, S., and Guestrin, C. why should trust you?" explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pp. 1135-1144, 2016.
Scott, M., Su-In, L., et A unified approach to interpret- ing model predictions. Advances in neural information processing systems, 30:4765- 4774, 2017.

---

Optimal Information Retention for Time-Series Explanations
A. Related Work
Deep Model Interpretability. Current efforts to enhance the interpretability of deep models primarily focus on exploring the relationship between inputs and predictions or making the prediction process more transparent (Räuker et al., 2023). Specifically, these research endeavors can be categorized into four primary directions: Employing visualization techniques to reveal the response relationship between feature maps any layer of the model and the input (Yosinski et al., 2015; Zeiler & Fergus, 2014). ii) Extracting rules or feature interactions, such as leveraging attention mechanisms to understand the role of each variable in predictions and identifying the main features driving the model (Choi et al., 2016; Qin et al., 2017). iii) Simulating the predictive behavior of deep models with interpretable models, such as approximating the decision surface of the model in the neighborhood of samples (Bhatt et al., 2020; Ribeiro et al., 2016; Guidotti et al., 2019). iv) Scoring the importance of input data and visualizing attribution maps (saliency maps or importance maps), which is currently the most active area of interpretability research, with representative methods like Integrated Gradients (IG) (Bachrens et al., 2010), Grad-CAN Crad (Selvaraju al., 2017), and SHAP (Lundberg, 2017). v) Mask-based methods are devoted to learning mask matrix M, and the Hadamard products of the input and M are represented as the salient features (Dabkowski & Gal, 2017; Chen et al., 2018; Fong et al., 2019). Jethani et al. point out that explanation methods suffer from computational efficiency, inaccuracy, or lack of faithfulness. The lack of robustness of the underlying black-box models, especially to the erasure of unimportant distractor features in the input is a key reason why certain attributions lack faithfulness (Bhalla et al., 2023). ROAR (Hooker et al., 2019) points out that many popular interpretability methods produce feature importance estimates that are no better than randomly assigned feature importance, and proposes an empirical measure of the approximate accuracy of feature importance estimates in deep neural networks. Although various interpretation methods have been applied to Computer Vision (CV) and Natural Language Processing (NLP), their application to time series often overlooks the modal heterogeneity. For instance, redundancy or incompleteness in time series interpretation may represent different temporal patterns.
Time Series Interpretability. Discriminative information in time series is often distributed across combinations of variables or multiple time steps. Designing specialized interpretation methods based on these characteristics of time series signals is the focus of current efforts (Doddaiah et al., 2022; Bento al., 2021: Guidotti al., 2020; Mujkanovicc et al., 2020). A representative approach employs attention maps as references for interpreting deep models of time series. For example, WinlT (Leung et al., 2021) employs attention maps as references for feature importance, while Dynamask (Crabbé & Van Der Schaar, 2021) applies regularization constraints to attention maps to achieve smooth temporal patterns. However, these methods use zero-padding as baseline for perturbation, which may generate -iittri-iitiiittionnn temporal patterns (Tonekaboni et al., 2020). The recent application of multiple instance learning to time series tasks has provided new nnighhs for interpretability, i.e., Early et al. propose multiple instance learning for locally explainable time series classification, which guarantees the intrinsic interpretability of the deep model without compromising the predictive performance. Chen et al. have explored the feasibility of introducing multiple instance learning into time series tasks from the perspective of information theory and weakly supervised learning. Another category of methods involves using surrogate models to interpret pre-trained black-box models, such as TIMEX (Queen et al., 2024) and TIMEX+ (Liu et al., 2024). However, TIMEX is limited by the out-of-distribution shifts of the surrogate model, while TIMEX++ overlooks the completeness of temporal patterns, thus failing to capture the full temporal dynamics.
Explanations of Information Theory. Information theory provides a novel perspective for explaining the complex dynamics of deep models. Representative methods include information bottleneck (Tishby & Zaslavsky, 2015), contrastive learning (Dosovitskiy et al., 2014; Khosla et al., 2020; Tian et al., 2020), and rate reduction (Yu et al., 2020; Chan al., 2022). Among these, the information bottleneck is used to interpret the learning process of deep models, describing it structurally as a trade-off between compact representation and information utility. Contrastive learning formalizes the pre-training objective as minimizing the distance between positive pairs in the embedding space while maximizing the distance between negative pairs, such as maximizing mutual information for positive samples and minimizing it for negative samples. Rate reduction posits that deep models are stacks of information compression and discretization (Yu al., 2023). Although these methods offer insights into the foundational dynamics of deep models, they lack intuitive interpretability at the data level, meaning they cannot directly guide the optimization of local explanations.
B. Technical Details from Section 2
We wish to reiterate the core contributions of our method from a technical perspective. Compared to heuristic interpretability methods, we first propose an optimal information retention principle from an information theory perspective and then
12

---

Optimal Information Retention for Time-Series Explanations
construct practical framework based on the principle. The principle provides optimization directions for explanations across three levels: fidelity, low redundancy, and high completeness. The specific contributions of the principle three-fold:
We align the prediction distributions of the original time series and explanations within the pre-trained black-box model to ensure the fidelity the explanation.
We employ variational methods learn mask matrix that follows Bernoulli distributions, masking task-irrelevant information to ensure low redundancy in the explanation.
We deconstruct the time series signal into positive and negative samples pairs, employing contrastive learning to ensure that positive samples contain task-relevant effective information while negative samples contain only irrelevant information, thereby ensuring the completeness of the explanation.
The second and third terms naturally derive objective functions from the explicit conditional mutual information in the optimal information retention principle, and we provide the corresponding proofs below.
B.1. Proof of Theorem
Minimum Redundant Information Retention. The optimal sub-features X should only contain the mutual information shared by T(X;Y) Y), and include as little information of X no relation to Y, as follows:
min I (X; Y).
(16)
We aim to derive from 7 (X;X") Y) the objective function for optimizing time-series explanations:
Liasst = DE[ (p(m x))]((mm))
Σ a log r
+ (1 pr,a) log Pt,d Pt.d
(17)
r
Proof.
I(XX"" Y)= I (prm) [ log
(x, xm 11)
p p(xx, xm ly) drdæ" plu)dy
pl v)p (x (8)
I (Jre / log
P (m
2. 1))
p (I,xx" ly) dxdx"") p(y)dy
P (I
ly)
I f log
p (mm (x,
p (æ xt" y) dxdx"
(18)
p (x0T
ly)
P (w x.
E Eppee..em.0) w)
log
p
v)
p (x" log
x y) (arm
Ep(x.w) Expam (2:0))
(v)
and we can view Epp(=1/2.)) log p(="'Iar p(u""1))
as the Kullback-Leibles divergence of p(am xr, v) and p(a |
y). So
T (X; XM Y) can be rewritten as:
XX; Y) = Ep(a.)) (pL((p("" (r.8))lp(="' 12))
(19)
where nm
( 17 and (19) can be redefined as:
(X; M Y) Ep(z.)) [DL(p((m (. w))]
(20)
So there are
min T (X; XM V) min DEL (p(m) x, v.llpmmm v)).
(21)
13

---

Optimal Information Retention for Time-Series Explanations
Since p(m y) is intractable, introduce a variational approximation g(m y) for this problem. We now proceed in two steps. First, we deal with the D minimization problem without y conditional constraint. Considering that any point mij E IRTx on M follows a Bernoulli distribution with value v = {0, 1), so the distribution of M can be assumed as:
((m) II Bern(r),
(22)
where the desired probability of is hyperparameter in practice. Similarly, parameterize the learnable multiple Bernoulli distribution of p(m z)) as:
pmm x)) II
(ern(
(23)
t,
where Pr.d denotes the learned probability of Fe,d is preserved.
Second, given the condition constraint the expected distribution q(m y) or p(m x. should implement the conditional probability with as a known term. We approximate the conditional constraint by considering the distributional consistency of (y. f)) as a complement to the conditional probability, as follows:
Las (f(x), (x")) = E (Ds (f(z))lf (x"")),,
(24)
where Das represents the Jensen-Shanan -hasnnnn (JS) divergence. And Las (f(z), f (x"") zo 0 the ideal case. Combined with the analysis the first step, the objective function derived from min / (X; M Y) can described as:
min E|Du =)]lg(m))), s.t. LLC (f(x), (x"))) x 0.
(25)
Since LJs ( (I*)) 2 0, so the constraint term combined in one function:
(in (p(m x)]])(m))) fLC (f(x), (x
(26)
where
E[DKL(p(m [)]l((m))]
apc log
Pt.d
(1 Pt,a) log
1 1 Pt.d
(27)
t.d
Furthermore, considering that the second term is the objective function corresponding to Semantic Information Retention, (26) can be further simplified to the minimization of the objective function:
F (Dl..n((( [))]g(()))]
= L log Pt.d + (1 - Pt,a) log
1 - Pt,d
(28)
1
t.d
Summary. We have shown the objective function (17) derived from min 7 Xx; X.M IY) This completes the proof of definition.
B.2. Proof of Theorem 2
Maximum Effective Information Retention. The optimal retention of sub-features X" should encompass as much task-relevant information from X as possible, minimizing the risk of information omission, follows:
min / (X;Y Y [
(29)
We aim to derive from 7 Y Xm) the objective function for optimizing time-series explanations:
exp (f (x:) (i:))
Lean
log
(30)
N
exp (f (a.) f (a:)) exp (f (z.) of (+,))
14

---

Optimal Information Retention for Time-Series Explanations
Proof.
J
xm (J, x
( y 2T)
I (X;Y )
log
P pa,y x²*) drdy p (x"m) dx
p | xm P (y xm
p (2, v. p(a)p (amM) p(z)p (u, xm)) (e, xm)
log
p (2, x"") dædydx"
xm
(rr ",
log P x
log
p (,,,,, 8, dodydr"
xm
p(z)p (y,
p(z) (zm
/ log
(z, 1,, x")
xx)x"")) P (2, y, x" drdydr" J xmm [ log p(x) (xm)) (x,, (") dxdx."
p(x)p (y: xm)
= I (X;Y, - (x;
(31)
and since XT as a sub-segment of X, cannot be directly used as input to f(.) for the different shapes. Therefore, X g(X"") is used as input quantify the conditional mutual information / (X; Y x assuming distribution (X) distribution(X). is the generator function, i.e., one layer of MIP. Then, there are
I (X;Y 1x") = (x: -1 (x; (xxxxx)) -1
(32)
To minimize I ( x"). we minimize the first term / (x;;,,)) while maximize the second term (Xx; X)) When
X serves as the input generated by X",, mutual information I ((; should ideally include only the information contained in ((XY), Otherwise, redundant information may introduced. Assuming X has no overlap with the portions of H(X Y), then I (X:Y.X)
(X;Y), which is constant independent of X.
For I (X; X), since X and X share the effective information in XT", they can naturally regarded as the anchor sample and the positive sample, respectively, contrastive learning. Therefore, we employ the contrastive loss including three maximize the mutual information between X and X:
exp (x:) (f:)) exp (x.) (a.))
Leam
log 1=1
(33)
N
exp (J (z.) (47)))
where žj denotes the negative sample, which can be generated by h(X X i... Gaussian noise imputation.
Summary. We have shown the objective function (30) derived from min (X;Y) X²). This completes the proof of definition.
C. Pseudocode of adapt-STE
Algorithm 1 adaap-STTE Estimator
Input: Logits 1, Forward temperature r/, Backward temperature nb, Updating rate 7), Standard deviation a, Upper bound of temperature Tmaz, Lower bound of temperature Tmin
2: Forward Pass:
3: Sample Gumbel noise: y Gumbel(0,) 1)
4: Update forward temperature: Tf clip (T o (l), Tmin: Tmax)
5: Compute relaxed logits for forward pass: softmax ( + +g)
6: Compute discrete mask: m = arg max(p)
7:
Backward Pass:
8:
Update backward temperature: 7, clip (Tu 17 (1), Tiins Tiaxx)
9: Compute relaxed logits for backward pass using same Gumbel noise sample: D² softmax ( + g)
Compute gradient using p² instead p
Perform backpropagation using the relaxed gradients
15

---

Optimal Information Retention for Time-Series Explanations
D. Exprimental Details
D.I. Description of Datasets
We evaluate our method on four carefully designed synthetic datasets with ground-truth annotations and four real-world datasets. The number of samples, length, dimension, number of categories and tasks for all datasets are summarized Table 3.
Synthetic Datasets. We employ four synthetic datasets with known ground-truth explanations proposed by Queen et al. (2024) to investigate the ability to identify interpretable time patterns.
FreqShapes is a dataset designed for discerning the frequency of anomalous signal occurrences. It comprises four classes: Class 0 presents a downward pulse every 10 steps, Category 1 presents an upward pulse every 10 steps, Class 2 presents a downward pulse every 17 steps, and Class 3 presents an upward pulse every 17 steps. The ground truth is explained by the positions of the ascending and descending peaks.
SeqComb-UV is a dataset tailored for distinguishing signal trends. It consists of four classes: Class O has no trend, Class includes subsequences with an upward trend, Class contains subsequences with downward trend, and Class simultaneously encompasses both upward and downward subsequences. The ground truth is explained by identifying the locations of the subsequences that exhibit trends.
Cem represents the multivariate version of SeqComb- UV. Subsequences with trends are randomly injected into the sensors. The ground truth also explained by pinpointing the positions of the subsequences that exhibit trends.
Low Var is a dataset aimed at identifying regions of low variance that change over time within multivariate time series samples. It is divided into four classes: Class O identifies discriminant subsequences with a mean of -1.5 on sensor O; Class 1 identifies discriminant subsequences with mean of 1.5 on sensor Class 2 identifies discriminant subsequences with a mean of -1.5 on sensor I; and Class 3 identifies discriminant subsequences with mean of 1.5 on sensor 1. The ground truth is explained by determining the locations of the discriminant subsequences.
Real-world Datasets. We employed four real- world datasets, among which only the ECG dataset has generated ground-trul truth explanations.
ECG (Moody & Mark, 2001) is dataset used for classifying cardiac arrhythmias. It includes three classes: normal readings (N), left bundle branch block beats (L), and right bundle branch block beats (R). The original dataset was segmented into 92,511 I samples by windowing, each containing 360 time steps. We selected these because it is known that the L and R diagnoses depend on the QRS interval, which will serve as our ground truth interpretation.
PAM (Reiss & Stricker, 2012) is a dataset utilized for discerning human activities. It encompasses 8 classes, each representing a different daily life activity. Every sample comprises 17 sensors with 600 consecutive observations, sampled at a frequency of 100 Hz, and the samples are roughly balanced across all 8 classes
Epilepsy (Andrzejak et al., 2001) is an electroencephalogram (EEG) dataset designed to distinguish epileptic seizures. It contains two categories: seizure and non-seizure. Each sample lasts for 1 second and is sampled at frequency of 178 Hz.
Boiler (Shohet et al., 2019) is a dataset from an industrial setting. The learning task involves detecting mechanical failures in the blowdown valves of boilers. This dataset is particularly challenging due to its large dimensionality ratio, unlike other datasets that contain more time steps than sensors.
D.2. Description of Baselines
IG. (Bachrens et al., 2010) Integrated Gradients (IG) utilizes the gradients model to derive attribution maps for explanation. This method compares gradients to a baseline value and performs Riemann integration. Although Integrated Gradients is a classical explanation method, it does not incorporate inductive biases specific to time series.
SGT + Grad. (Ismail et al., 2021) This method employs Saliency Guided Training. During the training process, features with low gradients are masked to guide the model to focus on more significant regions for prediction.
16

---

Optimal Information Retention for Time-Series Explanations
Table 3. The description of synthetic and real-world datasets.
DATASET
# OF SAMPLES
LENGTH
DIMENSION
CLASSES
TASK
FREQSHAPES
6,100
50
4
MULT-CLASSSIHCATION
SEQCOMB-UN
6,100
200
1
4
MULTTICLASSSIFICATION
SEV
6,100
200
4
MUT.TICCTASSSIIICATION
LowVAR
6,100
200
2
MULTI-CLASSIRICATION
ECG
92,511
360
1
ECG CLASSIFICATION
PAM
5,333
600
17
ACTION RECOGNITION
EPILEPSY
11,500
178
1
2
EEG CLASSIFICATION
BOILER 160,719
36
20
2
MECHANICAL FAULT DETECTION
Dynamask. (Crabbé & Van Der Schaar, 2021) This explainer is specifically constructed for time series and generates explanations using perturbation based iterations. The method iteratively occludes each sample to learn a continuous mask. The goal is to morph the input time series into a carefully determined baseline value, thereby highlighting important temporal patterns.
WinIT. (Leung al., 2021) Similar Dynamask, this method is designed for time series and learns mask matrix to achieve explanations. WinlT removes segments of the time series to measure the impact of features the final prediction value. It considers feature dependencies across time steps and uses a generative model to alleviate Out-of-Distribution (OOD) issues.
CoRTX. (Chuang et al., 2023) This method is an explainer that leverages contrastive learning approximate SHAP values. Originally developed for computer vision, we adopted a custom version implemented in aiming for the time series encoder and explanation generator work in concert.
TIMEX. (Queen et al., 2024) This method trains an interpretable surrogate to mimic the behavior of pre-trained time series model. It addresses the model fidelity issue by introducing model behavior consistency. preserves the relationships in the latent space induced by the pre-trained model and those induced by TIMEX, learning a discrete mask matrix to explain time series.
TIMEX++. (Liu et al., 2024) This method investigates from information-theoree perspective how existing interpretability methods may be affected by trivial solutions and distribution shift problems. To tackle these issues, TIMEX+ employs the information bottleneck principle to revise the objective function of TIMEX. This method currently the most competitive baseline approach.
D.3. Description of Metrics
As described by Crabbé & Van Der Schaar (2021), we employ AUP and AUR to evaluate the efficacy of explanation methods. As described above, let the binary mask matrix M E {0, 1] x the explanation of any black-box models for time series, and let Q E {0, 1) 7 be the ground-truth mask matrix. Qr.a then the feature of æt,a is salient; Otherwise Q then the feature of az task-irrelevant. We aim convert the mask into an estimator as follows:
Qua(t)= - {
if Mrd
(34)
else.
Consider the saliency index obtained by the truth value and explanation method separately:
A ((t,d) [1: [1: D] Qr, 1}, Á(r) = ((t,d) E [1 : x [1 D Q.a(T) 1}.
(35)
17

---

Optimal Information Retention for Time-Series Explanations
The precision and recall curves that map each threshold to precision and recall scores are as follows:
AA A(T)
P:: (0,
[0, 1]
|A(T)]
(36)
IA A(T)
R: (0,1) [0,1]==
|
So AUP and AUR can be derived by:
AUP P(r)dT,
(37)
AUR [( R(T)dT.
D.4. Description of Black-box Hyperparameters
In order to fairly compare the effectiivneess of interpretation methods, it is necessary to ensure that the parameters of the pre-trained black-box models are consistent. To this end, we refer to the parameter settings of Liu et al. (2024) as shown in Table 4.
Table 4. Training parameters for transformer-based predictors across all ground-truth and real-world datasets.
PARAMETUR
PREQSHAPE
SEOCoMB-UV
SEcCuMB-MV
Low VAR
ECG
PAM
EPILOPSY
BOILER
WAFER
PRUEZERRUGULAR
WATER
0.001
0.001
0.0005
0.001
0.002
0.001
0.0011
0.001
0.0001
0.0001
0.002
WEIGHT DECAY
0.1
0.01
0.001
0.01
0.001
0.001
0.001
0.001
0.001
0,001
0.001
100
500
NUM. LAYERS
de
16
64
128
32
64
72
16
32
16
16
64
DROPOUT
0.1
0.25
NORM. EMBEDDING
No
No
No
YES
YES
No
YES
No
No
Yas
D.5. Description of ORTE Hyperparameters
Table 5 summarizes the parameter settings for ORTE, where is utilized regulate the impact of contrastive learning on the interpretability effect. When is optimized, it preserves more complete interpretable temporal patterns, avoiding the loss valuable information. ,3 is employed to prevent overfitting and enhance generalization capabilities on test data. Y ensures the sparsity explanations while also serving as supplement to low redundancy. 7,, Tmaa and Tiin are hyperparameters corresponding to uTdp where we adopted the same parameter settings to ensure applicability to different datasets. All the experiments are performed on Ubuntu 18.04.6 LTS and GPU NVIDIA GeForce RTX 2080.
Table 5. Training parameters for ORTE across all ground-truth truth and real-world experiments.
PARAMETER
FREQSHAPE
SEQCOMB-US
SeqCOMB MY
LOwVAR
ECG
PAM
EPILEPSY
BOILER
LEARNING RATE
0.001
0.001
0.002
0.005
0.0005
0.0005
0.0005
0.0001
BATCH SIZE
64
64
64
64
16
32
32
32
WEIGHT DECAY
0.001
0.001
0.001
0.0001
0.0001
0.001
0.001
0.001
SCHEDULER
YES
YES
No
No
No
No
YES
YES
EPOCHS
50
100
100
100
50
50
0.5
0.5
0.5
0.5
0.5
0.1
0.5
0.5
17
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
3
3
3
3
3
3
Tminn
1
1
1
10.0
5.0
10.0
1.0
10.0
10.0
2,0
10.0
B
1.0
2.0
1.0
0.05
0.05
0.05
0.05
0.05
0.01
0.005
0.001
18

---

Optimal Information Retention for Time-Series Explanations
D.6. Description of classification Performance
Excellent classification performance is a necessary guarantee for evaluating the effectiveness of explanations. To objectively compare interpretation methods, employ the vanilla Transformer (Vaswani, 2017) as the black-box classifier. Table 6 summarizes the classification performance of the Transformer four synthetic datasets and four real-world datasets. The results show outstanding performance on all datasets except for dataset Boiler, with both AUP and AUR exceeding 0.9. The reason is that dataset Boiler is more challenging due to its larger dimension-to-length ratio, unlike the other datasets, which contain more time steps than sensors.
Table The performance of transformer-based predictors for time series classification.
DATASET
F1
AUPRC
AUROC
FREQSHAPES
0.9692:10.0055
0.9980=100000 0.993610.0026
SEQCOMB-UV
0.9412=0.0057
0.97891000034
0.9023+0.0011
SEVCOMB-MV
0.9270=0.0548
0.9445=100510
0.9792+:0.0144
LOWVAR
0.985411.0044
0.9008+000006
0.9992++0.0003
ECG
0.9211=0.0311
0.9463=100030
0.9009
PAM
0.887610.0064
0.9312=100004
0.9085=0.0015
EPILEPSY
0.9260:1.0014
0.93831:00009
0.9630=:0.0135
BOILER
0.8363=0.0118
0.821410.0263
0.8901
E. Additional Experiments on CNN and LSTM
We now study the applicability of ORTE across different black-box model architectures. Specifically, we conducted experiments on both convolutional neural networks (CNN) and long-short term memory (LSTTM) networks, which are specifically structured as follows:
CNN: 3 layer CNN MLP+ + meanpool.
LSTM: 3 layer bidirectional LSTM MLP mean of last hidden states.
The classification Performance of CNN and LSTM are summarized in Table 7 and Table 8, respectively.
Table 7. The performance of CNN-based predictors for time series classification.
DATASET
F1
AUPRC
AUROC
FREQSHAPES
0.9910:10.0011
0.9999:10.0002 0.99971100007
SBVCOMB-MV
0.9833-0.0171
0.998110.0024
0.9993+:00000
ECG
0.917810.0119
0.946610.0136
0.9682+:0.0522
Table The performance based predictors for time series classification.
DATASET
F1
AUPRC
AUROC
FREQSHAPES
0.9798=0.0141
0.997610.0018
0.9992-10.0077
SEVCOMB-MV
0.9332-0.1052
0.9473=0.1057 0.7907=10.159
0.9824:0.0348
ECG
0.8095:10.3622
0.8303:101189
We compared ORTE with five competitive baseline methods: IG, Dynamask, WinIT, TIMEX, and TIMEX++ Table 9 summarizes the experimental performance of ORTE for explaining CNN models on both univariate and multivariate synthetic datasets. Table 10 summarizes the experimental performance of ORTE for explaining CNN models on ECG datasets as real-world applications. Overall, the interpretation effectiveness of ORTE shows significant improvement compared to competitive baseline methods. Specifically, relative the currently most competitive interpretation method
19

---

Optimal Information Retention for Time-Series Explanations
TIMEX++, the AUP of ORTE increases by 36.86%, 0.31 %, and 13.62% datasets FreqShapes, Seqcomb-MV, and C, respectively. Furthermore, AUPRC respectively improved by 9.46%, 0.51% and 26.23%, which aligns with the trade-off between low redundancy and high completeness advocated in this paper.
Table 9. Explainer results with CNN predictor FREQSHAPES and SEQCOMB-M synthetic datasets.
FREQSHAPES AUP
SEQCOMB-MV
METHOD
AUPRC
AUR
AUPRC
AUP
AUR
IG
0.9905=0.0001
0.8777110.0009
0.705610.0017
0.597910.0027
0.8858110.004
0.229410.0013
DYNAMASK
0.2574=0.000S 0.4432++0.0032
0.5257+000015
0.0510+0.0016
0.0135 0.0005
WINIT
0.5321-0.009 0.6020+-0.0022
0.3966+-0.0017
0.5001
0.8324+000020
0.2259+000020
TIMEX
0.4966-10.0033
0.7916-000021
0.7016--0.0019
0.7670+000012
0.4689-0.0011
TIMEX+
0.9134=0.0014 0.6066+1.0001
0.7052
0.7822+400012
0.889610.0005
0.3434#0.0012
ORTE
0.9990=0.0001
0.8302+000013
0.7862++0.003 0.00971000001
0.802410.0005
0.3928=000020
Table 10. Explainer results with CNN predictor on ECG dataset.
ECG
METHOD
AUPRC
AUP
AUR
IG
0.4010+000010
0.3374+0.0012
0.530=00.0010
DYNAMASK
0.4090:0000100 0.7216=000027
0.1314=0.000
WINIT
0.3963=+00001 0.3292=1000200
0,3518=0.0012
TIMEX
0.6001::.0010 0.7451=000011
0.4161=000008
TIMEX
0.6020 0.7570=000001
0.4319=0.0012
ORTE
0.8490:10.0018
0.8601=10.016
0.4079=0.0012
Similarly, Table 11 summarizes the experimental performance of ORTE for explaining LSTM models on both univariate and multivariate synthetic datasets. Table 12 summarizes the experimental performance of ORTE for explaining LSTM models on ECG datasets as real-world applications. The experimental results for LSTM are also outstanding, especially on the multivariate dataset Seqcomb-MY, where AUPRC improved by 92%, AUP by 23.85% and AUR by 27 27.25%, respectively.
Table 11. Explainer results with LSTM predictor on FREQSHAPES and SEQCOMB-MY synthetic datasets.
FREQSHAPES AUP
SBOCOMB-MV AUP
METHOD
AUPRC
AUR
AUPRC
AUR
IG
0.9282=0.0016
0.7775:10.0010
0.692610.0011
0.236910.0020
0.51501100008
0.3211#0.0032
DYNAMASK
0.2290-0.0012 13342-1000037
0.5170+0.0013
0.2836+0.0021
0.636910.0047
0.1816+0.0015
WINIT
0.4171=0.0016 0.5106:10.00 0026
0.390910.0017
0.3515110.0014
0.654710.0026
0.3423#0.0021
TIMEX
119903-0.000 0.7887+10.00
0.79631000013
0.0091110.0017
0.1307-100002
0.47511000015
TIMEX+
0.0002
0.741310.0000
0.842810.0008
0.405211000038
0.680410.0052
0.351910.0021
ORTE
0.9990=0.0000
0.857340.0011
0.99931000003
0.7780-10.001
0.8427+0.0004
0.4478
Table 12, Explainer results with LSTM predictor on ECG dataset.
ECG
METHOD
AUPRC
AUP
AUR
IG
015037+0.0018
0.6129-0.00226
0.4022-0.0011
DYNAMASK
0.3730:10.0012 0.0099-0.00300
0.1102=0.0007
WINIT
0.302810.0013 0.3805=0.0022
0.4055=0.0009
TIMEX
0.605710.0018 0.6416-20.024
0.4436-0.0017
TIMEX
0.601240.0011 0.7432-0..001
0.4451=0.0008
ORTE
0.7372+000019
0.7817=0.0020
0.5672-0.0018
20

---

Optimal Information Retention for Time-Series Explanations
F. Computation Cost
evaluated the computation cost of interpretability methods on two real-world datasets (PAM and Epilepsy), as detailed in Table 13. Table 13 compares our method (ORTE) with four baseline approaches (IG, Dynamask, TIMEX, and TIMEX++) in terms of parameters (M). FLOPs (G), and Inference Runtime (s). Compared with TIMEX and Timex++, our inference time is slightly longer due the double-STE implement the regulation of discrete maps, but all three in the same rank, i.e., less than I second. IG and Dynamask require longer inference time, which is consistent with expectations since both methods operate recursively on single sample, while our method requires only one forward propagation.
Table 13. Comparison of computational costs.
Method
PAM
Epilepsy
Param(M) FLOPS(G) Inference runtime(s)
Param(M) FLOPS(G)
Inference runtime(s)
IG
8.7837 0.1862
23.9806 1.5245
Dynamask
458.4144 4 32.5353
979.5168 + 2.8082
TimeX 0.0620
6.5404
0.2475 = 0,1072
0.0169
0.3546
0,2545 0.1162
TimeX++ 2.7012
8.6530
0,2451 = 0,1013
0.0601
0,4487
0,2586 0.1176
ORTE
2.7097
6.7700
0,4914 0.0053
0.0645
0.3585
0.5858 0.0439
G. Contrast Experiments of adapt-STE
To further validate the effectiveness of adapt-STE, compared the performance of adapt-STE and STE four datasets: FreqShapes, SeqComb-UV, SeqComb-MV, and Low Var. As shown in Figure 6, adapt-STE significantly outperforms STE in metrics of AUPRC, AUP, and AUR. Particularly in AUP, adapt-STE demonstrates higher explanation accuracy, which is considered a more critical metric. Compared to STE, adapt-STE exhibits a lower AUR on dataset FreqShapes, which may be attributed to an imbalance between low redundancy and high completeness, resulting in the omission of some effective information.
2.0
Aipp STE STE
adberr STE
ARVT
572
S12
0.6
0.4
0.2
3.0
5.0
(c)
Figure Comparison performance between adapr-STE and STE, (a) Comparison performance AUPRC. (b) Comparison performance
of AUP. (c) Comparison performance of AUR.
H. Visual Comparison and Analysis
To facilitate a more intuitive comparison of various explanation methods, we have visualized the saliency maps these methods. We randomly selected three samples from four datasets, each accompanied by ground truth explanations, to qualitatively analyze the effectiveness the explanation methods through comparison with the ground truth. As shown in Figures 7-10, the results on datasets FreqShapes, SeqComb-UV, SegComb-VV and Low are visualized respectively, Each column represents a sample, and each row corresponds to explanation method, with the bottom row representing the ground truth explanation. In the figures, darker colors indicate more significant features, with the ground truth explanation being the ideal result. As shown in Figure 7, our method accurately highlights the downward spikes, which are fundamental to category judgment. In contrast, baseline methods suffer from issues of redundancy or incompleteness. For instance, TIMEX highlights too many time steps, causing irrelevant information to mix with explanatory temporal patterns, thereby failing to achieve effective interpretation. Conversely, Dynamask omits a significant amount of valid information, failing
21

---

Optimal Information Retention for Time-Series Explanations
meet the requirement for completeness and resulting in the absence of critical temporal patterns. Similarly, in Figure 8, method accurately highlights sub-segments with distinct trends. IG and TIMEX+ include excessive redundant background information, while Dynamask and TIMEX miss important trend segments. Figures 9 and 10 demonstrate the performance of the explanation methods on multivariate data. Notably, complex coupling relationships between variables make interpretation more challenging. Our method shows greater adaptability, further validating the advantages of the optimal information retention principle.
22

---

Optimal Information Retention for Time-Series Explanations
Integrated Gradients
h M .1
MVM
M/
2.-
4.-
-1.-
-000
Dynamask
W
MWM
-.5
-14
-10
M/
MM
TIMEX
-0.0
M
W
MWM
TIMEX+
-4.0
-100
M
W
Ours
-1.0
10.00
J004
Ground Truth
M
MMV
W/
-2-
-200
-20
Figure 7. Visualization of explanations on FregShapes dataset.
23

---

Optimal Information Retention for Time-Series Explanations
20
Integrated Gradients
Post Melle LMMt
Mmmarn
Ausu
Muma
Dynamask
Mmmmu
AmA Mwmn YYW
TIMEX
Mc/
Autu Mwma VYN
TIMEX+
Audu Mumu VYN
+1.00
+1.0
Ours
Aumu Mwma AyA
-1.0
-20
Ground Truth
10
Amd
Mwmn
14
+1.
Figure 8, Visualization of explanations on SeqComb-UV dataset.
24

---

Optimal Information Retention for Time-Series Explanations
MaLd 030
Integrated Gradients
Dynamask
TIMEX
TIMEX+
Ours
Ground Truth
Figure Visualization explanations on SeqComb-MV dataset.
25

---

Optimal Information Retention for Time-Series Explanations
WMA-MMM M Alittt
Integrated Gradients
Dynamask
WAKAANN My
Ayr mall deapllltt Mh
TIMEX
Wyfpe mial of ال P Ma
TIMEX+
MA My
Ours
Ground Truth
Figure 10, Visualization of explanations on Low dataset.
26


Reviews:
[{"review_id": "cjm4jxQfzf", "reviewer": "Reviewer_QJfM", "summary": "The authors address redundancy and completeness in time-series explanation methods by deriving an Optimal Information Retention principle from information theory, which optimizes explanations by minimizing redundancy while maximizing completeness. Based on this principle, they propose ORTE, a novel explanation framework that ensures informative and non-redundant explanations. They validate their approach through empirical evaluations on eight synthetic datasets and four real-world datasets, demonstrating its effectiveness over existing explanation methods.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comment:**\n\nWe sincerely appreciate Reviewer QJfM for the careful analysis of our work and valuable suggestions.\n\n- **Claims And Evidence:** 1. Occlusion alone does not fully capture redundancy and completeness. & 2. Insertion experiments should complement occlusion.\n\n  **Reply:** Thanks again for your advice. We believe the insertion experiments can serve as a beneficial supplement to our occlusion experiments. We completed the insertion experiments by gradually inserting the bottom $p$-percentage salient features, as shown in Figure R1 (https://anonymous.4open.science/r/orte/anonymous.pdf). Similar to the occlusion experiments, the imputation experiments compare our method (**ORTE**) with three baseline methods (**Dynamask**, **TIMEX**, and **TIMEX++**) and random insertion (**Random**) on three real-world datasets (**PAM**, **Epilepsy**, and **Boiler**). The results show that **ORTE** achieves a lower AUPROC when inserting the bottom 75% of salient features. As the insertion percentage increases, the predicted AUROC gradually improves. When the insertion ratio reaches 97.5%, ORTE attains the highest AUPROC. This indicates that the most interpretable or informative features are concentrated in the high-saliency features, further validating the manuscript's claims of low redundancy and high completeness. In addition, we notice that **Random** always maintains a high AUROC, which is because the random selection of features does not distinguish the salient or importance of features and contains informative time points or segments.\n\n- **Other Strengths And Weaknesses:** Other than Limited Experimental Evaluation for Real-World Datasets and Gaussian Noise Imputation for Negative Samples, there are no concerns.\n\n  **Reply:**\n\n  1) As replied in **Claims and Evidence**, the imputation experiments, serving as a complementary evaluation metric on real-world data, is able to jointly support the low redundancy and high completeness claims of this paper together with the occlusion experiments.\n  2) We investigated the impact of Gaussian noise intensity on negative sample constructions, as shown in Figure R2 (a) (https://anonymous.4open.science/r/orte/anonymous.pdf). We tested $ \\sigma = [0.1, 0.3, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0] $ eight groups of noise intensities on the multivariate synthetic dataset (**SeqComb-MV**). The results show that AUP is lower at $\\sigma \\leqslant 1.0 $, while AUPRC and AUR are higher, which indicates that insufficient noise intensity may be not beneficial for the separation of interpretation patterns from redundant information. When $1.5 \\leqslant \\sigma \\leqslant 3.0 $, AUP and AUPRC are higher, which indicates that the interpretation patterns of positive samples are effectively discriminable and do not produce OOD patterns. A value proximate to 2.5 is recommended for our experimental applications. And the more analysis of gaussian noise imputation for negative samples on other datasets will be included in the updated version.\n\n- **Questions For Authors:**  Comment on Figure 3. ... how do the results relate to redundancy and completeness?\n\n  **Reply:** For Figure 3, we stepwise occluded the bottom $p$-percentage salient features, following TIMEX and TIMEX ++. Ideally, the redundant features are assigned low saliency, and all the important features (i.e. completeness) are assigned high saliency. When the low saliency features are occluded, the prediction performance of the algorithm will be less affected. On the contrary, if the important features are identified as low saliency and are occluded, the prediction performance of the algorithm will be significantly reduced. On the one hand, this can reflect the requirements of redundancy and completeness in interpretation results, but also as pointed out in **Claims and Evidence**, due to the lack of ground-truth interpretation of real-world data, occlusion experiments still have the limitation of robustness. To this end, the imputation experiment can serve as a powerful supplement. Specifically, when the low saliency features are occluded or the high saliency features are inserted, the proposed algorithm can obtain higher prediction performance, which will further verify the claim of this paper.\n\n- **Essential References Not Discussed:** To strengthen this connection, the authors could discuss ROAR ([1], NeurIPS 2019) ... Referencing ROAR would further validate the effectiveness of ORTE's evaluation strategy.\n\n  **Reply:** Thanks for the friendly suggestion, and we will discuss the reference of ROAR in the updated version.\n\n- **Other Comments Or Suggestions:** 1. Typo. Line 208 (2.3) & 2. Hyperlink issue.\n\n  **Reply:** \n\n  1) We addressed this issue in the updated version.\n  2) We double-checked the hyperlinks to references and sections on multiple reading platforms.", "claims_and_evidence": "The paper proposes ORTE, an explanation framework designed to minimize redundancy and maximize completeness, verified through AURPC, AUP, and AUR on synthetic datasets with ground truth explanation labels. The evaluation methodology for synthetic datasets is well-structured and justified, as these metrics align with the paper’s theoretical objectives. Table 1 quantitatively supports the validity of ORTE in this controlled setting.\n\nFor real-world datasets, the paper employs stepwise occlusion experiments (Figure 3) to assess explanation validity. The use of AUROC as a primary metric makes sense in measuring how explanations influence model predictions. However, there are some limitations in the evaluation approach. \n1. Occlusion alone does not fully capture redundancy and completeness. The authors assume that a stable AUROC under extreme feature removal suggests explanation stability. However, if AUROC remains above 95% after occluding 99% of features, this may indicate a redundant explanation rather than a robust one. \n2. Insertion experiments should complement occlusion. The paper does not include stepwise insertion experiments, which could provide a more balanced assessment. Insertion tests could verify completeness by measuring how much AUROC recovers when adding back features, while also assessing redundancy by identifying how many features are actually necessary.", "methods_and_evaluation_criteria": "The validation strategy—conducted on eight synthetic datasets and four real-world datasets—is appropriate for the problem and application. The use of synthetic datasets with ground truth explanations allows for a well-controlled evaluation, and the metrics (AURPC, AUP, AUR) provide a reasonable measure of redundancy and completeness in this setting.\nHowever, for real-world datasets, the experiments are not sufficient to effectively demonstrate low redundancy and high completeness. The evaluation primarily relies on stepwise occlusion experiments, which, while informative, do not fully capture both aspects of explanation quality. As discussed in the Claims and Evidence section, integrating stepwise insertion experiments alongside occlusion would provide a more comprehensive assessment of redundancy and completeness.", "theoretical_claims": "The proofs for the two main criteria—minimum redundant retention and maximum effective retention—are provided in the supplementary material. Upon review, they appear to be correct with no immediate inconsistencies.", "experimental_designs_or_analyses": "As discussed in the Claims and Evidence section, integrating stepwise insertion experiments alongside occlusion would provide a more comprehensive assessment of redundancy and completeness, addressing current limitations in evaluating explanation quality.\nAdditionally, a minor potential improvement could be to analyze the sensitivity of the hyperparameter controlling the noise level in negative samples used in contrastive learning. Since the method applies Gaussian noise to generate negative samples, adjusting the noise level could have some influence on the contrastive learning process. While this is not a critical issue, a brief sensitivity analysis could help ensure the robustness of the approach.", "relation_to_broader_scientific_literature": "The paper builds on prior work in explainability by introducing ORTE, which optimizes redundancy minimization and completeness maximization. This aligns with information-theoretic approaches in explainability and contributes to time-series interpretability. While the Gaussian noise imputation technique for negative samples is not a major contribution, it serves as a simple way to avoid capturing OOD patterns, unlike the zero-padding approach mentioned in the related work section.", "essential_references_not_discussed": "The paper appropriately cites works related to explainability, information-theoretic approaches, and time-series interpretability. Its main motivation—the lack of a unified optimization criterion across explanation methods—is closely related to the evaluation of interpretability techniques. To strengthen this connection, the authors could discuss ROAR ([1], NeurIPS 2019), which introduced a systematic framework for assessing interpretability methods and formalized insertion and occlusion (deletion) as key evaluation techniques. Referencing ROAR would further validate the effectiveness of ORTE's evaluation strategy.\nReference:[1] Hooker, S., Erhan, D., Kindermans, P. J., & Kim, B. (2019). A benchmark for interpretability methods in deep neural networks. Advances in Neural Information Processing Systems, 32.", "other_strengths_and_weaknesses": "Another strength is clear Motivation and Theoretical Grounding. The paper effectively frames the problem of unifying redundancy minimization and completeness maximization in explainability, providing a principled approach grounded in information theory. Other than Limited Experimental Evaluation for Real-World Datasets and Gaussian Noise Imputation for Negative Samples, there are no concerns.", "other_comments_or_suggestions": "1. Typo. Line 208 (2.3)\n2. Hyperlink issue. The hyperlinks for references and sections did not work on my end. This may be a local issue, but please double-check to ensure proper linking.", "questions_for_authors": "- Comment on Figure 3. Beyond stating that the experimental results are better and more stable, what additional interpretations can be drawn from Figure 3? Specifically, how do the results relate to redundancy and completeness? A more detailed discussion on these aspects would enhance the interpretation of the figure.", "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": "AHIiSlhrxy", "reviewer": "Reviewer_voUp", "summary": "This paper introduces a novel approach to explainability in time-series deep learning models through an information-theoretic lens. The authors propose the \"Optimal Information Retention Principle\" which outlines three key criteria for high-quality explanations via information retention: ‘semantic’, ‘minimum redundant’, and ‘maximum effective’. Based on these they develop ORTE, a practical framework that learns binary masks to identify important patterns while filtering out redundant information. The framework incorporates contrastive learning to achieve a balance between low redundancy and high completeness in explanations. Experiments on both synthetic and real-world datasets demonstrate that ORTE outperforms existing SOTA explainability methods on similar metrics to those used by TIMEX/TIMEX++.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comment:**\n\nWe sincerely appreciate Reviewer voUp for offering valuable insights and recognizing of our work.\n\n- **Claims And Evidence:** \n\n  - C4. Practical utility: claim lacks detailed comparison of computational demands versus simpler approaches, which would be important for real-world applications. Introducing (even theoretical) estimation of time/memory complexity would be welcome.\n\n    **Reply:** We evaluated the computational requirements of interpretability methods on two real-world datasets (PAM and Epilepsy), as detailed in Table R1 (https://anonymous.4open.science/r/orte/anonymous.pdf). The computing infrastructure consisted of Ubuntu 18.04.6 LTS and an NVIDIA GeForce RTX 2080. Table R1 compares our method (**ORTE**) with four baseline approaches (**IG**, **Dynamask**, **TIMEX**, and **TIMEX++**) in terms of parameters (M), FLOPs (G), and Inference Runtime (s). Compared with **TIMEX** and **Timex++**, our inference time is slightly longer due to the *double*-STE to implement the regulation of discrete maps, but all three are in the same rank, i.e., less than 1 second.  **IG** and **Dynamask** require longer inference time, which is consistent with expectations since both methods operate recursively on a single sample, while our method requires only one forward propagation.\n\n  - C5. Real-world evaluations: while the occlusion experiments on real-world datasets without ground truth (Figure 3) show ORTE maintains high prediction AUROC, this is only a proxy measure and doesn't definitively prove the explanations are correctly identifying the truly important features. Other work has introduced evaluations by consulting subject-matter experts and comparing their interpretation of variables of importance with those generated by the proposed framework.\n\n    **Reply:** Thanks for your advice. We consulted clinical experts and analyzed real-world **ECG** data, as shown in Figure R3 (https://anonymous.4open.science/r/orte/anonymous.pdf). Figure R3 compares our method with **IG**, **TIMEX++**, and **Ground Truth** (QRS complex waves). Although **IG** accurately locates some points of the QRS, more information is missed compared with the Ground-Truth. While **TIMEX++** highlights the salient features, it also mixes the redundant features, which makes it difficult to understand the prediction of the algorithm. Our method maintains less redundancy while highlighting more complete salient features, which verifies the optimal information retention principle proposed in this paper. Notably, clinical experts identified a critical annotation discrepancy: the reference Ground Truth inadvertently incorporated proximal P-wave components (non-QRS elements) at leftward leads. This observation suggests that the benchmarks may require narrower physiological bounds, which further supports that our method may provide new insight for the interpretation of real data.\n\n- **Other Strengths And Weaknesses:**\n\n  - W1. The real-world evaluations as previously mentioned.\n\n    **Reply:** For the evaluation of real-world data, on the one hand, we followed your friendly advice and consulted clinical experts as a supplement to the explanation evaluation, as mentioned in C5 reply of **Claims And Evidence**. On the other hand, we followed the advice of reviewer QJfM and supplemented the imputation experiments, as detailed in the reply to the Claims And Evidence section of reviewer QJfM. \n\n  - W2. No significant investigation into edge-cases like dealing with irregularly sampled or missing data.\n\n    **Reply:** We investigated edge-case of missing data, as shown in Table R2 (https://anonymous.4open.science/r/orte/anonymous.pdf). We compare the proposed method **ORTE** with the strongest baseline **TIMEX++** on SeqComb-MV data. We set three scenarios including random loss of 5% and 15% of data points and random loss one variable, respectively. The results show that our method performs well on AUPRC, AUP and AUR, which is consistent with the claims of the manuscript.\n\n- **Questions For Authors:**  \n\n  - Q1. How does the computational complexity of ORTE compare to methods like IG or TIMEX++? \n\n    **Reply:** Please refer to C4 reply of **Claims And Evidence**.\n\n  - Q2. How sensitive is the method to the choice of hyperparameters, particularly those in the contrastive learning component?\n\n     **Reply:** We investigated the hyperparameter choice of contrastive learning (i.e., $ \\alpha $), as shown in Figure R2(b) (https://anonymous.4open.science/r/orte/anonymous.pdf).  When $0.1 \\leqslant \\alpha \\leqslant 13 $, AUPRC and AUP slowly rise and gradually plateau, while AUR gradually rises. The AUP will decrease significantly when $\\alpha $ is larger, that is, the explanation fails. A value proximate to 10 is recommended for our experimental applications.\n\n- **Essential References Not Discussed:**\n\n  **Reply:** Thanks. We will supplement and analyze these references in the updated version.", "claims_and_evidence": "Overall, the core technical claims about ORTE's performance improvements are well-supported, but some broader claims about its practical advantages and generalizability would benefit from additional evidence.\n\nWell-supported:\nC1. Performance superiority: The claim that ORTE outperforms existing methods is convincingly supported by comprehensive experiments across multiple datasets. The quantitative results in Tables 1-2 show clear improvements over competitive baselines, particularly on AUPRC, AUP, and AUR metrics.\nC2. Theoretical foundation: The claim that information theory can provide optimization criteria for explanations is supported through detailed mathematical derivations and proofs in the paper and appendices.\nC3. Adaptability: The claim that ORTE works across different model architectures is demonstrated through additional experiments on CNNs and LSTMs in the appendix.\n\nLess convincing evidence:\nC4. Practical utility: claim lacks detailed comparison of computational demands versus simpler approaches, which would be important for real-world applications. Introducing (even theoretical) estimation of time/memory complexity would be welcome.\nC5. Real-world evaluations: while the occlusion experiments on real-world datasets without ground truth (Figure 3) show ORTE maintains high prediction AUROC, this is only a proxy measure and doesn't definitively prove the explanations are correctly identifying the truly important features. Other work has introduced evaluations by consulting subject-matter experts and comparing their interpretation of variables of importance with those generated by the proposed framework.", "methods_and_evaluation_criteria": "The proposed method of using binary masks to identify important regions via a transformer-based mask generator -> contrastive separator -> predictive distribution aligner makes sense after the introduction of the three information retention criteria.\n\nThe synthetic evaluations with ground truth are standard for time series interpretability and show the performance of the model against a wide range of competing methods. Although the real-world evaluation is somewhat lacking, the approach does not seem too far off. There is some evaluation missing when considering how features are being masked - it seems the authors are simply masking the bottom x-percentile but this does not take into account temporal dependencies within each dimension of a multivariate time-series - they should consider masking until a 1-std change or until the end of the series.", "theoretical_claims": "I reviewed the proofs provided in Appendices B.1 and B.2, which derive the objective functions from the information-theoretic principles. The mathematical derivations appear sound and follow established information theory principles.\nNo errors were identified in these derivations, and the theoretical foundation appears solid.", "experimental_designs_or_analyses": "Again, the synthetic experimental design is comprehensive and appropriate:\n- The use of synthetic datasets with ground truth explanations (FreqShapes, SeqComb-UV, SeqComb-MV, LowVar) provides a clear baseline for evaluating explanation quality.\n- The ablation studies effectively isolate the contribution of each component in the framework.\n- The visualizations provide qualitative evidence that complements the quantitative metrics.\n\nHowever, the experimental design could be improved:\n- While  real-world datasets from different domains were included (ECG, PAM, Epilepsy, Boiler) the experiments themselves were not rigorously compared.\n- The authors could provide runtime/computational complexity comparisons across methods\n- A hyperparameter sensitivity analysis could be completed", "relation_to_broader_scientific_literature": "The paper situates itself well within existing TS explainability literature, particularly at the intersection of time-series analysis and information theory. The authors appropriately acknowledge:\n- Prior work on local explanations in time series (Dynamask, TIMEX, WinIT)\n- Information-theoretic approaches in deep learning (Information Bottleneck, contrastive learning)\n- The challenges specific to time-series explainability (temporal dependencies, out-of-distribution issues)\nThe work advances the field by providing a unified information-theoretic framework specifically designed for time-series data, addressing the limitations of methods that were originally designed for images or text.", "essential_references_not_discussed": "The work does address most relevant work, however it excludes recent advances on applying multiple instance learning to time series tasks. Early et. al. (2024) [https://arxiv.org/pdf/2311.10049] and Chen et. al. (2024) [https://arxiv.org/abs/2405.03140] provide an introduction to these methods and how they provide inherent interpretability for multivariate time series classification problems from an information-theoretic perspective.\n\nEarlier work by Tonekebani et. al. (2020) [https://arxiv.org/abs/2003.02821]  and Ismail et. al. (2020) [https://arxiv.org/abs/2010.13924] which helped set the stage for time series interpretability and provides other synthetic and real-world evaluations for improved experimentation could further support this work.", "other_strengths_and_weaknesses": "S1. The paper provides a unique and theoretically sound unified optimization framework for TS explainability\nS2. The adapt-STE technique is a novel contribution that could be useful for other binary mask learning problems.\nS3. The synthetic evaluations show significant performance improvements compared with previous SOTA.\n\nW1. The real-world evaluations as previously mentioned.\nW2. No significant investigation into edge-cases like dealing with irregularly sampled or missing data.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Q1. How does the computational complexity of ORTE compare to methods like IG or TIMEX++?\nQ2. How sensitive is the method to the choice of hyperparameters, particularly those in the contrastive learning component?", "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": "IL1QADsS2y", "reviewer": "Reviewer_w93i", "summary": "This paper proposes an explanation method called \"ORTE\" (the title of the paper) which uses the Optimal Information Retention principle to construct explanations for time series models. Specifically, given a time series classifier, they propose a mask generating model to produce a binary mask for each input, such that three important information theoretic conditions (Criteria 2.1 - 2.3) hold. These criteria essentially require (1) that the masked input be sufficient for prediction; (2) the masked input has the lowest possible mutual information with the full input; and (3) the complement of the masked input must contain as little information about the label as possible. Experiments demonstrate that the proposed explanations are close to the ground truth explanations, handily beating other methods in this area.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comment:**\n\nWe express our sincere gratitude to Reviewer w93i for providing comprehensive review, insightful perspectives, and thought-provoking questions.\n\n- **Claims And Evidence:** However the claim that \" We achieve ... on real-world datasets (Figure 3).\n\n  **Reply:** Thank you for your advice. We will revise the presentation to faithfully reflect the experimental conclusions.\n\n- **Methods And Evaluation Criteria:** However, the paper misses a discussion of the interpretability of the masks... A lack of discussion / experiments / visualizations on this aspect is a critical drawback of this paper.\n\n  **Reply:** Thank you again for your constructive suggestions, which will be very helpful for a discussion on interpretation and expert cognition. We evaluate the meaning of mask-based explanations with expert explanations from quantitative and qualitative perspectives, respectively. \n\n  1) For synthetic data, we followed the same experimental setup as **TIMEX** and **TIMEX++** with pre-defined Ground-Truth explanations. Quantitative analysis of the consistency between salient features and Ground-Truth can accurately evaluate the interpretation effect. For visualization of synthetic data FreqShapes explanation, as shown in Figure R4 (https://anonymous.4open.science/r/orte/anonymous.pdf). Compared with the strongest baseline **TIMEX++**, our method maintains stronger consistency with the Ground-Truth. Likewise, we provided corresponding visualizations and discussions in the supplementary material of the manuscript. \n  2)  For annotated real-world data, ECG data has expert-predefined Ground-Truth to refer to. Table 2 of the manuscript provides a quantitative evaluation. In order to further explore explain the results and human domain expert cognitive consistency, we visualize our method and the interpretation of the baseline method, as shown in Figure R3 (https://anonymous.4open.science/r/orte/anonymous.pdf). The Figure R3 compares our method with **IG**, **TIMEX++**, and **Ground Truth** (QRS complex waves).  In addition, we invited clinicians to analyze the interpretation results. The professional physician proposed that the Ground Truth in the figure may be mixed with the P wave on the left that does not belong to the QRS wave, and the more accurate Ground Truth should be more narrow, which further verified that our method may provide an insight for the interpretation of real data. \n  3) For the real-world data without annotations, on the one hand, we followed TIMEX and TIMEX++, progressively occluding the saliency features at the bottom of the $p$-percentage. Ideally, the redundant features are assigned low saliency, and all the important features (i.e., completeness) are assigned high saliency. When the low saliency features are occluded, the prediction performance of the algorithm will be less affected. On the other hand, we followed the suggestion of reviewer QJfM and supplemented the imputation experiment to support the requirements of redundancy and completeness in the interpretation results, as detailed in the reply to the Claims And Evidence section of reviewer QJfM.\n\n- **Other Strengths And Weaknesses:** the trade-offs of the mask continuity.\n\n  **Reply:** \n\n  1) For mask continuity, we follow the TIMEX experimental setup, we also investigated the influence of continuity, as shown in Figure R2 (c) (https://anonymous.4open.science/r/orte/anonymous.pdf). We tested AUPRC, AUP, and AUR on SeqComb-MV data. The results showed that continuity did not affect the algorithm significantly, so we chose a consistent setting.\n\n  2) For the sparse mask, Figure R2 (d) (https://anonymous.4open.science/r/orte/anonymous.pdf) investigated the sensitivity. In the interval $[0.001, 0.05]$, the interpretation of the algorithm is relatively stable. However, there will be mode collapse when the sparsity is too large, which is consistent with our expectation that effective features will be missed when the sparsity is too large.\n\n- **Questions For Authors:**  \n\n  - Q1. the human-interpretability of the mask explanations, and the impact of the \"mask continuity\" parameter?\n\n    **Reply:** \n\n    1. For the **human-interpretability of the mask explanations**, please refer to the reply of **Methods And Evaluation Criteria**.\n    2. For the **impact of the \"mask continuity\" parameter**, please refer to the reply of **Other Strengths And Weaknesses**. \n\n  - Q2. the incomplete literature survey, and how the proposed method compares to the literature on masking explanations?\n\n    **Reply:** We will further clarify our literature survey, and **TIMEX** and **TIMEX++** as our baselines, following masking explanations, both are based on mask learning saliency distributions. And the other baseline method **Dynamask**, is also based on mask learning. In the updated version we will add a more extensive discussion, including time series and other general methods as you mentioned in Essential References Not Discussed.", "claims_and_evidence": "The claims made in the paper are mostly well supported by experiments. However the claim that \" We achieve state-of-the-art performance on eight synthetic and real-world time series datasets compared to the latest competitive baselines\" is a bit misleading: as the method performs on-par with others (Timex++) on experiments on real-world datasets (Figure 3).", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria are standard and makes sense for the problem at hand. However, the paper misses a discussion of the interpretability of the masks: while the proposed masks are broadly faithful to the underlying model, it is unclear whether they are also interpretable or meaningful to a human domain expert. It would be good to know for example, if using this method results in greater insights than using other competing methods (e.g.: timex++)? A lack of discussion / experiments / visualizations on this aspect is a critical drawback of this paper.", "theoretical_claims": "I did not check the correctness of the proofs, as they are in the supplementary material.", "experimental_designs_or_analyses": "The experimental design is sound and valid.", "relation_to_broader_scientific_literature": "This paper adds onto the literature on time-series explainability using feature attribution, and proposes a method to improve explanation faithfulness. In particular, this improves upon another SOTA time-series explanation method, called timex++, but augmenting the mask generation procedure to include more constraints, such as the minimality of the information between the masked and the full input; and the sparsity of the masks.", "essential_references_not_discussed": "The paper misses discussion of key literature in the broader explainability literature that performs masking in a manner similar to this paper. For example:\n\n1. Ruth Fong et al. \"Understanding deep networks via extremal perturbations and smooth masks\". CVPR 2019\n2. Dabkowski and  Gal. \"Real time image saliency for black box classifiers\".NeurIPS 2017\n3. Chen et al, \"Learning to explain: An information-theoretic perspective on model interpretation\", ICML 2018\n4. Yoon et al, \"Invase: Instance-wise variable selection using neural networks\", ICLR 2019\n5. Jethani et al., \"Have we learned to explain?: How interpretability methods can learn to encode predictions in their interpretations\", AISTATS 2021\n6. Bhalla et al, \"Discriminative Feature Attributions: Bridging Post Hoc Explainability and Inherent Interpretability\", NeurIPS 2023\n\nAll of these papers introduce methods for generating per-sample masks, and include discussions of the sufficiency, sparsity criteria similar to this paper. The missing discussion, and the lack of contextualization of the current methods against this larger literature is another critical drawback of this paper.", "other_strengths_and_weaknesses": "Additional Weaknesses:\n- The paper introduces 5 criteria in total (criteria 2.1-2.3) + mask sparsity + mask continuity, where the latter two are described more as implementation details. It is unclear which of these criteria is sufficient for good empirical performance in the results shown. For example, what are the trade-offs of the mask continuity and sparsity constraints on the results? The paper lacks a discussion of these aspects.\n\nAdditional Strengths:\n- The paper demonstrates good empirical performance, especially on recovering known explanations in synthetic settings.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Could the authors please clarify the concerns raised regarding:\n1. the human-interpretability of the mask explanations, and the impact of the \"mask continuity\" parameter?\n2. the incomplete literature survey, and how the proposed method compares to the literature on masking explanations?", "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": "Dewc8dAWLU", "reviewer": "Reviewer_Szo4", "summary": "This paper proposes the Optimal Information Retention Principle to improve explanations of deep models for time-series data by minimizing redundancy and maximizing completeness using conditional mutual information. The authors develop ORTE, a framework that learns a binary mask to filter irrelevant patterns while preserving key temporal features. ORTE leverages contrastive learning for precise filtering and ensures stable optimization. Experiments demonstrate superior accuracy and completeness over existing methods.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comment:**\n\nWe extend our sincere appreciation to Reviewer Szo4 for providing valuable feedback and acknowledgment of our research.\n\n- **Theoretical Claims:** This paper has three criteria: Semantic Information Retention, Minimum Redundant Information Retention, and Maximum Effective Information Retention. In the appendix, the authors provide proofs of the equivalence between these criteria and the implementation loss functions. The proof of Theorem 1 appears to be well-structured, but this reviewer is uncertain about the proof of Theorem 2, particularly the relationship between Equations 32 and 33.\n\n  **Reply:** For Equations 32 to 33, it is described how to derive positive and negative samples and the corresponding loss function in contrastive learning. Specifically, we learn a mask matrix $M$ that can separate the effective features, and mask $X$ to get $X^m$. However, $X^m$ cannot be directly used as a positive sample, as this would introduce too many 0 values and cause OOD. To do this, we use an MLP to map $X^m$ to $\\widehat{X}$. Conversely, the inverse mask filters out redundant features, which can be used as negative samples that are not relevant to the task. The positive sample $\\widehat{X}$ and the negative sample $\\widehat{X}^-$ thus constructed can be used for contrastive loss, namly, Equations 33.\n\n- **Other Strengths And Weaknesses:** \n\n  - In Figure 1, the description is not precise. Since the overlapping area represents mutual information, the green area should correspond to $H(X)$.\n\n    **Reply:** Thanks for your advice. We will fix this description in an updated version.\n\n  - The implementation of Criterion 2.3 is unclear. In line 167, the assumption states: \"Assuming $\\widehat{X}$ has no overlap with the portions of $H(X | Y )$\". However, the statement, \"For the second term, since X and $\\widehat{X}$ share effective information in $X_m$ , they can naturally regarded as the anchor samples and the positive samples, respectively, in contrastive learning.\" contradicts this assumption.\n\n    **Reply:** For $\\widehat{X}$ can be considered as the product of $X^m$ learned by MLP, while $X^m$ is considered as the valid feature retained after occlusion. Ideally, $H(X^m)$ should correspond to $I(X; Y)$ consistent. When the information of $Y$ is excluded, the remaining redundant information of $X$ can be considered as irrelevant to $\\widehat{X}$, so \"no overlap with the portions of $H(X | Y)$\". Furthermore, $\\widehat{X}$and $X^m$ have information consistency with respect to the label $Y$ that $X$ is oriented to, which is also the motivation for constructing positive samples. So $\\widehat{X}$ can be used for contrastive learning from $X^m$. Accordingly, $X$ can be regarded as the anchor of contrastive learning.", "claims_and_evidence": "1. This paper employs the Optimal Information Retention Principle to guide the identification of explanatory temporal patterns within time series, including Semantic Information Retention, Minimum Redundant Information Retention, and Maximum Effective Information Retention. In Section 2, three criteria are used to support these claims.\n\n2. The experiments and framework validate the contribution claims made in the introduction.", "methods_and_evaluation_criteria": "1. The method is well-organized and clearly presented, making it easy for readers to follow its main idea. In Figure 2, the Optimal Information Retention Principle is aligned with three components.\n\n2. This paper follows TimeX's evaluation protocol, using four synthetic datasets and four real-world datasets as benchmarks. It employs Area Under Precision (AUP), Area Under Recall (AUR), and Area Under the Precision-Recall Curve (AUPRC) as evaluation metrics. There are no issues with this section.", "theoretical_claims": "This paper has three criteria: Semantic Information Retention, Minimum Redundant Information Retention, and Maximum Effective Information Retention. In the appendix, the authors provide proofs of the equivalence between these criteria and the implementation loss functions. The proof of Theorem 1 appears to be well-structured, but this reviewer is uncertain about the proof of Theorem 2, particularly the relationship between Equations 32 and 33.", "experimental_designs_or_analyses": "The experimental protocols are well-designed, incorporating quantitative analysis, visualization, and ablation studies. The results and analyses are consistent and well-aligned.", "relation_to_broader_scientific_literature": "This paper proposes the Optimal Information Retention Principle to guide the design of explainable methods in the time series domain and introduces the ORTE framework. Experimental results demonstrate the effectiveness of this approach.", "essential_references_not_discussed": "There is no other reference papers from this reviewer.", "other_strengths_and_weaknesses": "Strength:\n1. This paper introduces a new framework, ORTE, for explainable time series models based on three criteria. The authors validate the effectiveness of these criteria through both theoretical and empirical analyses.\n2. The paper is well-organized.\n\nWeakness:\n1. In Figure 1, the description is not precise. Since the overlapping area represents mutual information, the green area should correspond to  $H(X)$.  \n2 . The implementation of Criterion 2.3 is unclear. In line 167, the assumption states: \"Assuming $\\hat{X}$ has no overlap with the portions of H(X | Y )\".   However, the statement, \"For the second term, since X and $\\hat{X}$ share effective information in $X_m$ , they can naturally regarded as the anchor samples and the positive samples, respectively, in contrastive learning.\"  contradicts this assumption.", "other_comments_or_suggestions": "Please find the comments above.", "questions_for_authors": "Please find the comments above.", "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": {"ethical_review_concerns": "There is no ethical review concerns in this paper."}}]

Author rebuttal:

Meta-review: The authors propose an approach based on the optimal information retention principle from information theory to address redundancy and completeness in time-series explanation. This approach minimizes redundancy while maximizing completeness when generating explanations.

The reviewers appreciated the work and the "clear motivation" of the work and its "theoretical grounding". While the reviewers generally liked the experimental design, they recommended additional sensitivity analysis (see Reviewer QJfM's response to the rebuttal). I think including such an analysis would be insightful. The reviewers have also pointed out missing references to recent advances on applying multiple instance learning to time series tasks and broader explainability literature that performs masking. I think the authors should expand the related work to include these works.
Decision: Accept (poster)