Uncertainty-aware Graph-based Hyperspectral Image Classification
Authors: Linlin Yu
Yifei Lou
Feng Chen
Venue: ICLR 2024 poster (2024)
Keywords: Uncertainty Quantification
Graph
Hyperspectral Image Classification
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Hyperspectral imaging (HSI) technology captures spectral information across a broad wavelength range, providing richer pixel features compared to traditional color images with only three channels. Although pixel classification in HSI  has been extensively studied, especially using graph convolution neural networks (GCNs), quantifying epistemic and aleatoric uncertainties associated with the HSI classification (HSIC) results remains an unexplored area. These two uncertainties are effective for out-of-distribution (OOD) and misclassification detection, respectively. In this paper, we adapt two advanced uncertainty quantification models, evidential GCNs (EGCN) and graph posterior networks (GPN), designed for node classifications in graphs, into the realm of HSIC. We first reveal theoretically that a popular uncertainty cross-entropy (UCE) loss function is insufficient to produce good epistemic uncertainty when learning EGCNs. To mitigate the limitations, we propose two regularization terms. One leverages the inherent property of HSI data where each feature vector is a linear combination of the spectra signatures of the confounding materials, while the other is the total variation (TV) regularization to enforce the spatial smoothness of the evidence with edge-preserving. We demonstrate the effectiveness of the proposed regularization terms on both EGCN and GPN on three real-world HSIC datasets for OOD and misclassification detection tasks. The code is available at GitHub.

Published as a conference paper at ICLR 2024
UNCERTAINTY-AWARY GRAPH-BASED HYPERSPEC- TRAL IMAGE CLASSIFICATION
Linlin Yu', Yifei You? Feng Chen'
'Department of Computer Science. University of Texas at Dallas {linlin. yu, feng. chen
dallas d edu *Department of Mathematics and School of Data Sciences and Society, University of North Carolina at Chapel Hill yf yflou@unc, edu
ABSTRACT
Hyperspectral imaging (HSI) technology captures spectral information across a broad wavelength range, providing richer pixel features compared to traditional color images with only three channels. Although pixel classification in HSI has been extensively studied, especially using graph convolution neural networks (GCNs), quantifying epistemic and aleatoric uncertainties associated with the HSI classification (HSIC) results remains an unexplored area. These two uncertainties are effective for out-of-distributio (OOD) and misclassification detection, respec- tively. In this paper, we adapt two advanced uncertainty quantification models, evidential GCNs (EGCN) and graph posterior networks (GPN), designed for node classifications in graphs, into the realm of HSIC. We first reveal theoretically that popular uncertainty cross-entropy (UCE) loss function is insufficient produce good epistemic uncertainty when learning EGCNs. To mitigate the limitations, we propose two regularization terms. One leverages the inherent property of HSI data where each feature vector is a linear combination of the spectra signatures of the confounding materials, while the other is the total variation (TV) regularization enforce the spatial smoothness of the evidence with edge-preserving We demon- strate the effectiveness of the proposed regularization terms on both EGCN and GPN on three real-world HSIC datasets for OOD and misclassification detection tasks. The code is available at GitHut
INTRODUCTION
Hyperspectral (HS) imaging widely used in various real-world applications including atmospheric science (Saleem et al., 2020). food processing (Ayaz et al. 2020). and forestry (Khan et al.) (020), benefiting from rich spectral information measured at individual pixels. Unlike human eyes which possess only three color receptors sensitive to blue, green, and red channels, HS data provides a wide spectrum of light (visible and near-infrared range) for every pixel in the scene, which enables more faithful classification results compared to traditional classification using color images. As result, hyperspectral image classification (HSIC) attracts considerable research interests (Chen et al., 2014]
Ahmad et al. 2017) [Hong et al. 2018 Ahmad et al. 2021). Specifically, graph convolution neural
network (GCN) (Kipf & Welling (017) has found extensive use in HSIC (Shahraki & Prasad 2018
Qin et al.) 2018 Wan et al.] 2020 Hong al.) 2020) due to its ability to effectively model the interdependency among pixels (especially when they are far away).
However, there is limited work related to predictive uncertainty quantification for HSIC. For ex- ample, it is not practical to assume that all categories (materials) in the scene are known and have available samples for model training. In such scenarios. the model is expected to have the capability to know what they do not know, which can be measured by epistemic uncertainty from a probabilistic view (uncertainty of model parameters due to limited training data). On the other hand, pixels may be misclassified due to various factors, such as environmental noise, material similarity, and atmo- desirable for training model to identify the unknown what they do not
spheric effects. Thus, know, which can be measured by aleatoric uncertainty (uncertainty due to randomness). Overall, it is necessary to quantify these two uncertainties to ensure the reliability of HSIC models.
betps://githube. httt::///itthbb.ccc//iiiin//////
nty-aware aware git

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The epistemic and aleatoric uncertainties can be used to detect outtof--ditrrbuttinn (OOD) pixels that belong to unknown materials and detect pixels that are misclassified to the wrong categories, respec- tively. OOD detection in HSIC performs in in-distribution (ID) classification and OOD detection si- multaneously, which is different from anomaly detection, as the latter only involves detecting pixels whose spectral characteristics deviate significantly from surrounding ature has found that epistemic uncertainty is the most effective for OOD detection, while alcatoric uncertainty is most effective for misclassification detection (Zhao et al. 2020 Stadler al. 2021).
background pixels. Liter-
basph models for HSIC construct a graph by regarding each pixel as a node and the inter- dependency among nodes is defined by an adjacency matrix. As result, the nodes on the graph are dependent on each other. In contrast to extensive literature for independent inputs (Lakshmi- narayanan et al. 2017 Gal & Ghahramani 2016 Charpentier et al., 2022), uncertainty estimation for semi-supervised node classification on graph with dependent inputs is more complex and thus less explored (Abdar et al. 202T). Notably, two primary investigations have been conducted employ- ing deterministic methodologies. One is the evidential graph convolutional network (EGCN) (Zhao ct al., 2020) that extended the evidential neural network (ENN) (Sensoy et al.) 2018) on images (independent inputs) to graph data (dependent inputs) with GCNs and graph-based kernel Dirichlet estimation. Throughout the paper, we refer to this model as GKDE or EGCN for brevity. The other is graph posterior network (GPN) (Stadler al., (021) that adapted the posterior network (PN) (Char- pentier et al. 2020) together with an evidence propagation through the graph nodes. Both GKDE and GPN predict the conjugate prior distribution of categorical distribution, i.e. Dirichlet distribu- tion at each node, and incorporate the uncertainty-crow crosss enosse (UCE) loss (Biloš et al.. 2019) in the overall optimization problem to train the model parameters.
However, the UCE loss function has limitations in effectively learning uncertainty quantification models. First, the UCE-bbasd learning models tend to peak the Dirichlet distribution, thus becoming overly concentrated on the predictive classes in the simplex (feasible) space composed of the class- probability vectors (Bengs et al. 2022). This drawback can be alleviated by introducing entropy- based regularization to encourage the predicted Dirichlet distributions to be uniform (Charpentier eet al. 2020; Stadler et al.) 2021). Second, it is known empirically that learning models based on UCE alone do not produce accurate epistemic uncertainty for OOD detection, which can be aided by additional regularization terms, e.g., the aforementioned GKDE (Zhao et al., 2020). However, show in our experiments both GKDE and GPN do not have satisfactory results in HSIC.
In this work, we consider the uncertainty quantification task for graph-based hyperspectral image classification. Our Contributions are summarized as follows. First, we characterize scenarios when minimizing the UCE loss fails to provide accurate epistemic uncertainty estimation. In par- ticular, minimizing the UCE loss does not help an EGCN to learn embeddings that are capable of mapping OOD nodes into the detectable region near the decision boundary. Second, we propose a multidimensional uncertainty estimation framework for HSIC. To the best of our knowledge, this is pioneer work in discussing the uncertainty estimation on the graph-based HSIC models. Third, we introduce a poysics-guider unmixing- the UCE loss when quantifying epistemic uncertainty. Here, we assume that OOD pixels are mostly composed of an unknown material and the UR term is the reconstruction squared loss for decom- posing into the ID materials and the OOD material. Fourth, we adopt the TV term propagate predicted evidence along the decision boundary (not across), thus preserving spatial edges between ID and OOD nodes. Finally, we present extensive empirical experiments demonstrate the effec- tiveness of the proposed regularization terms on both EGCN and GPN, using three real-world HSIC datasets for OOD and misclassification detection, in comparison to five baselines.
regularization (UR) to address the shortcomings of
PRELIMINARY
2.1
GRAPH-BASED HYPERSPECTRAL IMAGE CLASSIFICATION (HSIC)
HSIC aims to assign unique label to each pixel based its spectral and spatial properties. The input HS data can be represented as X [xx _,',,² x x (HW)) E [HHW)) where B is the number of spectral bands (feature dimension) and H x W is the spatial dimension. Letting N = HW, we stack the 2D spatial domain to vector, and hence each pixel is associated with a feature vector x' E R B Vi E [N]. For classification purposes, each pixel i has a class label y e [C] associated with specific constituent material, where C is the number of classes known priori.
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The graph-based HSIC technique (Qin al. 2020) builds graph, in which each vertex corresponds
to a pixel in the 2D spatial domain and the weighted adjacency matrix A is calculated based on similarities between node-level features: i.c.,
Aij = exp(-d(x' )x)) Vi,j e
e [N].
(1)
where d(x',x) x²) is the Euclidean distance (or cosine similarity) between vertices and J. and is tuned to optimize how similar two nodes are. Suppose the graph is defined as G(V, E, X, YL), where V {1.... N} is a ground set of nodes, E C x V is a ground set of edges, X [x:', x² ..... x E IR N is the node-level feature matrix, xx RR is the feature vector of node i, YL [y'li E [} E R44 is the label for the training node set L C V. and y E [C] is the label for node The GCN-based HSIC method (Hong et al. 2020 formulated as [p').ev = s(A, X; 0), where p² is the probability vector of node i and [()) is standard GCN function that depends on the adjacency matrix A, the data matrix X, and a set of network parameters, denoted by 0.
2.2 EVIDENTIAL GRAPH CONVOLUTIONAL NETWORKS FOR NODE CLASSIFICATION
An evidential GCN (EGCN) (Zhao et al., 2020) takes graph 9 as INPUT and predicts evidence
vector e' = [ei,-... -.. .ec] for each node i as OUTPUT: Je' = [(A, X; 0), where ea is a measure of the amount of support collected form the training labels YL in favor of node to be classified to the class c. EGCN is the same as classical GCN, except that the activation function (c.g., exponential or ReLUU) of the output layer is unbounded, outputting an evidence vector, instead of the softmax function outputting class probabilities. The evidence vector can quantify predictive uncertainty through a well-defined theoretical framework called subjective logic (SL) (Jsang 2018
More specifically, a multinomial opinion = (b, u) in SL can be defined as:
Cc and C for e c e.1, 5
b.
(2)
S
where = Ib....... be]" represents the beliefs of the C' classes, u is the uncertainty mass repre- senting the vacuity of evidence, and $ = -e=i(h (e. + 1). It is straightforward that be 2 0, u 2 0, and Ee be + u- 1. A multinomial opinion can be equivalently represented by Dirich- ~ Dir(a)), where = [p,, ,Pe] is a probability vector of C classes and = [a1, acc are called concentration parameters with og = ee 1. The class label y", prob- ability vector p', and the evidence vector e' for node i have the following probabilistic relations:
let distribution:
y' y'~ Cat(p"), p¹ ~ Dir(p") la'), a' = e' + 1, [e'] = (A,X; 0)
(3)
The expected class probability is equal to the mean of the Dirichlet distribution, i.c. p =
s in the sense that S can also be defined by S Σ ae. Based evidential theory, lack of evidence, e.g.. "I don't know," can be expressed by close-to-one vacuity u (or a uniform Dirichlet).
An EGCN is trained based on the uncertainty cross-entropy (UCE) loss function, defined by
U 0) Ep--pipp [- log (y Ip')
(4)
which can be interpreted as the expectation of the standard cross-entropy loss with respect to the dis- tribution of class probabilities: p² ~ Dir(p")a") Alternatively, Stadler et al.) (2021) proposed a new network architecture (as opposed to classical GCN architecture), namely graph posterior networks (GPN), to predict node-level Dirichlet distributions. Specifically, GPN consists of three modules: multilayer perceptron (MLP) layers for node-level feature embedding, a normalizing flow module to estimate node-level densities in the embedded space, and a personalized page rank propagation layer Gasteiger et al. 2018) smooth the concentration parameters among neighboring nodes.
2.3 UNCERTAINTY QUANTIFICATION
Aleatoric uncertainty is the uncertainty in the class prediction, which is measured by the entropy of categorical distribution(Mallinnn et al. 2017), i.e. walen (Ca((p))) or confidence (Charpentier max.Pe- It exhibits higher values when the categorical distribution is flat.
et al. 2020), i.e., u In contrast, epistemic uncertainty is the uncertainty on categorical distribution and can be measured by the total evidence count, i.e. 2 pis evidential uncertainty (Josang et al., 2018). When the distribution of categorical distribution, which is the Dirichlet distribution in the evidential-based models, is spread out, the epistemic uncertainty is high. Aleatoric uncertainty is proven to be effective for detecting misclassifications while epistemic often used to identify OOD samples (Zhao et al. 2020
C/S. which is referred to vacuity from the viewpoint of
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3 UNCERTAINTY-NWARE REGULARIZED LEARNING
3.1 LIMITATIONS OF UCE AND EXISTING REGULARIZATION TECHNIQUES
Without the loss of generality, we focus on binary classification tasks throughout this section for theoretical analysis; the generalization to multiple classes can be analyzed similarly tolCollins al.) (202)) and Kristiadi et al.] (2020). A typical EGCN architecture has several graph convolutional (GC) layers followed by one MLP layer (Zhao et al. 2020). GC layers produce node-level embed- dings to capture the graph dependency among the nodes in the sense that nodes that are neighbors in the graph are more likely to be spatial neighbors in the embedded space, denoted by D R. Specifically for homophily graphs (Ma et al. 2021). GC layers can generate embeddings that can separate different classes. Note that the three HSIC datasets used in our experiments are indeed homophily graphs, as discussed in Appendix B The MLP layer in EGCN helps to reduce the di- mensions of the embedded space while producing node-level evidence. We demonstrate that the MLP layer learned based on UCE fails to produce accurate evidence predictions, even in the ideal case where the GC layers can produce perfectly separable node embeddings. Let z' E EDD denote the embedded vector of node i E V. The MLP layer for node-level evidence prediction can be formulated as:
e(z: 0) == (e+(2: e), e- (z: 0)] = [o(w"z z + b). o- b)].
(5)
where 0- = {w, 6}. W E RD b E R. and (-)) is the activation function (e.g. ReLU and exponential) that outputs evidence values. We start with the lower and upper bounds for MLP-based ENNs.
Proposition Suppose z E IDD a point in the embedded space and y E {-1,+1} is its binary class label. An MLP-based ENN has the lower and upper bounds for the UCE loss:
1
5 [e_y(z: 0]1+1 11
S O))
(6)
Ciyy (z, +1
ey(2; 0)
where ez( 8) is the evidence of classs y, a<z 0) = e(z; + 1, and [-] denotes the ceiling operator. If ENN can predict y correctly: y(e+ 0) e- 0)) 0, have tighter upper
bound:
r+1
UCE(((Z), y: S (CE(((()) y: 0)
c,(2;
where for the ReLU activation function and 1 for the exponential activation function.
Please refer to Appendix A for the proof of Proposition I Note that the upper bound in is tight
under the universal approximation theorem (Pinkus) 1999). Specifically, the optimal parameter 0*
that minimizes the UCE on a training set has the property: (2; 0*) Do and e (2; 0 + 0, as
demonstrated in Lemma in Appendix I Therefore, we have
r ++1
r ++
1
lim
UCE((, 0) lim
cy(220)
Cy cx(z;
ey(x:0) +00 ey (z; 0) ey 0) +1
e, 0) +r+1 0.
lim
0y(x: +00 (c,(z; 0) (2c)(2;0))
Next, we establish in Theorem I that the optimal solution when minimizing the upper bound UCE(a(Z)) y; 0) defined in Equation 7 with exponential activation function a(-) has closed- form expression that is equivalent to the optimal solution of linear discriminative analysis (LDA) under certain assumptions.
Theorem 1.
We assume that (i) feature vectors belonging to classes {+1} follow Gaussian distri- butions with the same covariance matrix and the means =Pμ, respectively, i.e., P(z, y) = IP = +1)N(2; u, E) + PPyy
(2) ps, E), with P(y = +1) Ply -1) = 0.5; (äi
the optimal solutions 0" that minimize E(za)) (CE can linearly separate both classes: (2( 0) > (z 0), V(2, y). Let o() be the exponential function. The optimal solution
0- = (w*,b*) is the same as the optimal solution of LDA, iee,
w" = Σ-11p 4 and b* = 0.
(8)
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Theorem I has several important implications when the classes are separable. First, the MLP layer in EGCN learned based on UCE has the same objective as in LDA: Finding a projection that maximizes the separation between the projected class means with small variance within each class. Unfortunately, as illustrated in Fig. | this objective does not help learn a projec- tion that maps OOD data points to the grey region of low evidence near the decision boundary,
where GCNs can effectively detect OODs: ey (y 0*) 0 (or equivalently w* 2 b* 0).
For any far-away OOD data point 2 = 8.2, where ō - 20 and (z,3) PPZ.,) v) an ID data point, the predicted evidence approaches too (or equivalently the epistemic uncertainty approaches O): (z; 0*) exp(y(w**) br)) exp(yw* öz) (exp(yw** 2))² + oo, when z² z'Σ + O. given that b* = 0 and exp(y(w""z b*)) cxp(yw* 2) > 1. Second, the evidence predic- tions for the testing data points are not influenced by the distance between the class means, it not factor in w and b" Third, Fig.[]shows that we can identify the light-blue and light-grey OOD regions of different char- acteristics in the feature space of based on the projec- tion w* EE'Ep "r. In particular, the learned MLP layer predicts higher evidence for OOD nodes in the light-blue region than the one of ID nodes; and predicts evidence similar to those of ID data points for OOD data points in the light-grey region. The learned MLP can only predict small evidence for OOD points in the small light-grey re- gion near the decision boundary: w* T b" O.
(x) en"
Figure 1: Two-separable-class case: The grey region in the feature space de- tectable OOD region by one-layer MLP- based ENN Iearned using UCE. The light- blue and light-green OOD regions are not detectable by ENN.
We remark on several assumptions in Theorem I First, the assumption the Gaussian means μ and -u al- ways be true by translating the origin of the feature space to the middle point of two centers step. Second, we assumed the same covariance matrix Σ for the two Gaussians to obtain an analytical solution in Eq. (8) For different covariance matrices, the optimal so- lution is non-identical to that of LDA. Third, we assume that the classes are linearly separable so that the MLP layer can be defined in Eq. ( The linear separability has been assumed in OOD-related theoretical analysis such as Ahuja et al. (202T). For the non-separable case, the MIP layer is defined
a preprocessing
(x)x) = eulzohy >
[e+ (z; 0), e-)ee 0)] [o(w] z + bi), a (w, z + b2)]. where the weight and bias parameters for predicting the evidence values of the two classes are different. As demonstrated in Fig. [ the grey, light-green, and light-biue regions have more complex
e(z; 0)
Two-non-see senn----parraal-llase The grey, light-green, and light-blue regions are the same as those Fig [ In the orange region, the predicted evidence values of both classes are larger than 1.
Figure 2:
shapes compared to the separable case in Fig. [ The OOD points that can be detected by the
MLP layer are within the grey region: {zle, (z: 0) < - 0) < 1}. Further, there is an orange region for the non-separable case: {zle, (z: 0) > (,e-v(z; 0) >> 1}. in which the evidence values for both classes are larger than 1. Our theoretical results on EGCN may not be generalizable to GPN. GPN predicts evidence values based on density estimation in the embedded space instead of MLP layers as used in EGCN.
We demonstrate that minimizing the UCE loss does not help to learn the MLP layer to map the OOD regions (e.g., night-green and light-green OOD regions Figs. T and 2 into the detectable OOD region of EGCN near the decision boundary. Zhao et al. (2020) proposed to use the KL divergence-basee regularization term: EtEl (Lirii (a"), Dir p a to enforce the closeness between à' and a Vi E V, where o' a pre-computed teacher based on graph-kernel distance. However, this term assumes that OOD test nodes are far away in terms of graph-based distance from the training (ID) nodes compared to ID test nodes. This assumption is not always valid based on our empirical performance of the GKDE teacher on three HSIC datasets in Appendix E.5
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3.2 UNMIXING AND WVIDENCE-BASED UNCERTAINTY REGULARIZATIONS
Due to the limited spatial resolution of HSI sensors, it is conceivable that each pixel in HSI data may contain a combination of materials, and hence it is desirable decompose a single pixel into the proportions of constituent materials (a.k.a. abundance) (Tordache et al. 2012). We assume that there exist C' pure materials (a.k.a. endmembers) in the scene, each with the corresponding signature m. E IR 8 C E {1. ... C}. A matrix formed by all of these signatures is called mixing matrix, denoted by M = (.,..... me] E RBxC, The abundance map obtained by the abundance coeff- cients of all the pixels can be represented by matrix V E RCCN, We adopt a linear mixing model in which the spectral mcasurement at each pixel is a linear combination of the endmembers, i.c.
m'me + n'
xt
where of is the abundance coefficient for the c-th material at the ith pixel and ny' denotes a noise term. Denote v² = [oi................ VE].. It is typical to assume Ee=1 1/ = 1, as each abundance vec- tor resides within the probability simplex. We adopt the linear mixing model (9) for its simplicity. There are more complicated nonlinear models by taking into account endmember-wise scaling fac- tors (Drumetz et al., 20166. spectral variability (Hong al. 2018). and illumination-inen induued vari- ability (Drumetz et al. 2019). Decomposing the HS data X into collection of reference spectral signatures M with associated abundance matrix V is referred to as hyperspectral unmixing.
Unmixing-based Regularization (UR). HSIC is related to hyperspectral unmixing in that the in- distribution (ID) classes are associated with the C' known endmembers. Under the OOD detection setting where OOD classes are associated with unknown materials, we can consider linear mixing model, where the signatures of the ID materials, {m,,... } are given. We assume that OOD nodes are associated with the same unknown material that is denoted by m.. The hyperspectral unmixing problem can be formulated as
min mo 2
Σ
|| MM' - u.
(10)
We propose to use the belief's b'(0) and the vacuity u (0) (the epistemic uncertainty measure) to approximate the abundance coefficients vi of ID materials and the abundance coefficient vy of the OOD material, respectively. The rationale of such approximations, b' (0) ~ (0) and (0) z (0), is threefold. First, the sum-to-one property on beliefs and vacuity: Ee b + u² = 1, is aligned with the one abundance coefficients: Σ 1/ + vy 1. Second, the vacuity u² for ID node should be close to zero, and hence the beliefs are analogous to class probabilities (Jsang) 2018), which can be used approximate the abundance coefficients (Chen al., 2023). Third, the vacuity for an OOD node is close to one and its belief is close to zero, implying that the abundance coefficient ut (0) should be close to one and v² (0) be close to Using the approximations, we turn the unmixing problem I0 into an unmixing regularization (UR) term,
min UR(m:: 0) ;= EiEy || Mb (0) (0) m
(11)
where b', uV (Vi E V) can be derived by the evidence e0) SA 0), or e for brevity. Minimizing the UR term encourage high vacuity for OOD nodes and low vacuity for ID ones. Given 0, there is a closed-fon foom solution for the optimal mo, i.c.,
m Eiey (0) Mb
(12)
(ie((u,(0))=
Please refer to Appendix A.3 for more details. Using the definitions of b =
C+
e
e
u =
CO we rewrite UR (m 0) with respect to evidence e, i.e.,
EE me C- C-
Cm C+
UR(e) = Eiey ||
(13)
Proposition 2. Assume the linear model (9) holds without noise, the gradient descent for minimizing the UR regularization increases the predicted evidence of ground-truth class for ID instances and decrease the total evidence for OOD instances with the corresponding pure material contained in the pixel. Formally, we have

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(a) For an instance (a', .y) with feature matrix x' my and y E {1,.... , C} is the ground truth label, one has
aUR(E)
(14)
Oel
(b) For an OOD instance instance (x* y²) with x² m and y² = 0 [I,...,cC
aUR(e) de 2 O.
(15)
We present two desired properties of the UR term in Proposition 2 Part (a) is consistent with minimizing the UCE loss for ID nodes, aiming to predict high class wise evidence for ground truth class. This often results in an increased total evidence for ID nodes during training iterations. Part (b) implies decrease in total evidence for OOD samples when minimizing the UR term, resulting in a higher vacuity score, which provides additional information beyond the UCE loss. It is the inherent physical characteristics of hyperspectral data that implicitly help distinguish OOD and ID. Specifically, each pixel in hyperspectral image contains a spectrum, which is mixture of the spectra of all materials present in that pixel. ID and OOD pixels naturally contain different materials. Note that Proposition is agnostic to model architectures, i.c., the UR term can be applied to any uncertainty quantification architectures.
Evidence-based Total Variation Regularization (TV). The total variation (TV) regulariza- tion (Iordache et al. 2012) was applied to the abundance coefficients to enforce the spatial smooth- ness while preserving edges for hyperspectral unmixing. As the graph does not incorporate spatial information, we propose the use of TVV on the node-level vacuity value, which is inversely propor- tional to the Dirichlet level strengths (or equivalently total evidence). To define the discrete TV regularization, we represent 2D image of size H X W as vector via linear indexing, i.e., ((h ((h-1)) 1) + m))thh component denotes the location at (h,m). Define two matrices Da,, D, to be finite forward difference operators with periodic boundary conditions in the horizontal and vertical directions, respectively. Then the discrete form of the (anisotropic) TV norm is defined by
TV(u) || + DDD
(16)
Regularized Learning.
Putting together, we formulate the regularized learning objective as, REL (UCE(A', 0)) + A2 X2UR((0 TVT(())))),
C(O, m.)
Σ
(17)
where R(0) refers to the model (GKDE or GPN)-specific regularization term and Å₁ A2, As are hyperparameters. For GPN, R(0) Eier eer ENT(Dir(p''a')) The GKDE regularization term can be found in the last paragraph of Section 3.1 The TV term is applied on the vacuity score u()). The last two terms only require node features and are applied the whole graph V. The model parameter 0 and m., in UR term can be optimized alternatively: closed-form solution for m. in 12 and gradient descent to update the model parameters
EXPERIMENT
4.1
EXPERIMENT SETUP
Datasets We use three HSIC datasets for evaluation: the University of Pavia (UP), the University of Houston (UH), and the Kennedy Space Center (KSC). For train/(validation test) split, we adopt the public challenge split for UH (Debes et al. 2014), the same split for UP as (Hong et al. 2020). and random split for KSC with 20 nodes for training. For validation/test split, we use 0.2/0.8. The number of disjoint train/ maiin/validaion/tess samples selected from each class used for all the experimental results is presented in Appendix B
Competing Schemes We consider two state-of-the-at uncertainty quantification backbones de- signed for graph data: EGCN (Zhao et al. 2020) and GPN (Stadler et al.] 2021). For EGCN, we include GKDE regularization by default. Softmax-GCN (Kipf & Welling (017)) is a classic GCN for semi-supervised node classification and uses the softmax the last activation layer. We

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use the entropy as the uncertainty score (Hendrycks & Gimpel 2016 Though this paper fo- cuses on OOD detection, we include three anomaly detection models: TLRSR (Wang et al. 2022). RGAE (Fan et al. 202T), and TRDFTVAD (Feng al., 2023) basclines. We use the features of all the nodes on the graph for anomaly detection and regard the OOD nodes as anomalies. The detailed settings along with parameter tuning presented in Appendix [ We evaluate the perfor- mance via the area under the Receiver Operating Characteristic (AUROC) curve and the area under the Precision-Recal (AUPR) curve with epistemic uncertainty score in OOD detection tasks and aleatoric uncertainty score in misclassification detection tasks. We report the mean and standard deviation over five random trials in each table.
4.2 RESULTS
Misclassification detection. The problem of misclassification detection is to identify whether given prediction is misclassified or not with estimated uncertainty scores. A misclassified prediction given a positive label (i.c..1,, while correct prediction is given negative label (i.e., 0). Table I shows the misclassification detection results where the best performance over all the models is high- lighted in bold. We observe that softmax- GCN is decent on misclassification detection, indicating that misclassified nodes tend to have predicted class probabilities spread out across ID categories and entropy can capture reasonable aleatoric uncertainty for deterministic softmax models. Besides, our proposed uncertainty quantification frameworks show comparable results of the misclassification detection task compared to softmax -CNN on UP and UH. while better on KSC.
Table 1: AUROC and AUPR for the misclassification detection.
UP
KSC
dalasel
AUROC
AUPR
AUROC
AUPR
AUROC
AUPR
softmax- GCN
78.95-11.18
47.58:10.78
89.22.10.25
1.
89.22.10.25
$2.85 1.62
EGCN
77.37:10.61
47.78:10.44
87.9810.50
06.18:10.82
90.18:10.32
14,38
EGCN UR
1.3
47.16:10.52
88.47.10.56
76.77.11155
90.21:10.38
13.51
EGCN UR- TV (Ours)
78.83400.85
48.73-00.49
87.96400.69
15.76=1151
90.37:10.44
55.09=
GPN
03.39
44.38:0.00
80.06::0.95
67.24:11.92
18.13+13.82
.4.6877.09
GPN- UR
73.58:20.36
45.02::0.42
81.08:1.05
67.25::1.24
62.38:26.78
65.28:6617
TV (Ours)
73.35
47.99:12.37
83.361
67.57.1883
78.23:1..8
$3.08.
The bold numbers are the best results over all models. The underlined numbers are the best results within the same model type.
OOD detection. OOD detection aims to determine whether a given example is out-of-distribbtion (OOD) or in-distribution (ID) by assigning an estimated uncertainty score. An OOD example is labeled as 1, while an ID example is labeled as O. The experiments of the OOD detection are conducted using the left-out class setting, which aligns with the procedure in /Zhao et al.) (2020) and Stadler et al. (202T). Note that we exclude the left-out class from the training set but retain the nodes belonging to this class in the graph. Specifically, we randomly select one category as the OOD class, while the remaining categories are considered ID classes. Within each dataset, we create four random configurations, designating one distinct class OOD in each configuration. The weighted average, factoring in the number of test OOD nodes for every dataset, is displayed in Table Please refer to Appendix E.5 for more detailed settings and results. In Table 2 the bold numbers denote the best results over all types of architectures (i.e. GCN, EGCN, and GPN), while the underlined numbers are the best results within each type if not highlighted.
Table 2: AUROC and AUPR for the OOD Detection.
UP
UH
KSC
AUROC
AUPR
AUROC
AUPR
AUROC
AUPR
Soltmax-GCN
57.04
163413.29
26.78:1263
19.1820.57
77.12400.65
$4.18=1.2
RGAE
17.22-0.a.
2..1=n.a.
52.51-n..
10.594n..
69.62-n..
34.T0=tn.a.
TLRSR
74.03-t0..a.
20.11:in...
48.95tn...
6.24-nn...
$8.14in...
9.84:tna.
TRDETVAD
68.70-11.28
17.72-20.5
n.a.t..a..
n.a.bo..a.
30.4742.338
13.13=0.69
EGCN
8..27
45.50.10.65
88.64
1.76
89.29110.13
70.17-10.55
EGCN
90.43-1-0.1
46.06-20.31
99.81-00.56 56
43.25422.5
49.00-0.3
1..81=1.111
BGCN- UR- (Ours)))
9157-+0.11
06.44
90.69=+0.46
46.77=2.90
92.21=00.42
72.13=1.63
GPN
82.82-13.21
20.96112.50
82.16:11.25
46. 36.30
39.66:1..82
59. 09.30
GPN
93.63-10.62
1.87
04.75
19.57
88.40-10.80
62.01.1.81
GPN ./R TV (Ours)
94.55-10.23
51.74
87.29-L-1.04
$2.02
88.78-1-..11
61.11-1111
bold numbers are best results over models. The underlined numbers are best within same model type. n.a.
means either model or metric not applicable.
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We observe that proposed uncertainty quantification framework with both EGCN and GPN outperforms softmax-GCN and three anomaly detection baselines. First, this suggests that softmax entropy cannot effectively capture the epistemic uncertainty, resulting in poor performance on OOD detection. Second, anomaly detection techniques aimed at identifying pixels with abnormal features struggle to recognize OOD samples, possibly because abnormal features do not consistently align with features identifying OOD samples. In addition, the GPN backbone performs the best on UP, while EGCN performs best on UH and KSC. This difference in performance may be attributed the GKDE regularization in the EGCN model, which appears to have a stronger influence on UH and KSC, crucial for the OOD detection performance of the EGCN model. As experimental evidence, we utilize the GKDE regularization for OOD detection and achieve ROC values of 87.24% and 87.2% on UH and KSC respectively, while 69. 69.88% on UP.
Discussions. We emphasize the contributions of the proposed UR and TV terms based on Table T and [ The key findings are as follows. For misclassification detection, both UR and TV demonstrate ability to enhance the performance of EGCN and GPN. Specifically, UR improves performance in 5 out of cases, while TV improves performance in out of 6 cases. In terms of OOD detection, both UR and TV exhibit significant improvements across all datasets. GPN-UR achieves enhancements of up to 10% in AUROC and 7% in AUPR observed on the UP dataset. Subsequently, we apply TV regularization to EGCN-UR and GPN-UR, showing 1% increase in AUROC and a 3.1 1% increase in AUPR on the UP dataset compared to GPN-UR. A comprehensive ablation study provided in Appendix E.I and Table 12
It is also worth noting that AUROC and AUPR values are not always consistent. For example, on UH, GPN-UR-TV has higher AUPR but a lower AUROC than EGCN-UR-TV. AUROC and AUPR offer different perspectives for measuring the quality of a ranking on data points to separate positives and negatives. Davis & Goadrich (2006) pointed out that algorithms optimized for AUROC are not guaranteed to optimize AUPR, and vice versa. Yuan et al. (203)) also reported similar empirical observations for classification tasks. A low AUROC but a high AUPR for GPN-UR-TV indicates that GPN produces more true positives among the top-ranked nodes than EGCN, while EGCN can separate true positives and negatives better than GPN among the bottom-ranked nodes.
To further illustrate the above ob- servation, Figure 3 displays example curves of AUROC and AUPR on the UP dataset, with "shadows" selected as the OOD class. While TLRSR and RGAE exhibit impressive AUROC performance (over 93%), their AUPR outcomes are notably poor (below 12%), in contrast the PR of our proposed framework (over 93%). A high AUROC with an extremely low AUPR for balanced dataset indi- cates that the model tends to produce tremendous false positive errors. For example, most nodes have similar high predicted uncertainty scores, making them indistinguishable between ID and OOD.
To0e VELL
E0Ck 18:0 call
REDOLE 10
Figure 3: OOD detection on UP with "shadows" as OOD class.
CONCLUSION
We propose a graph-based uncertainty quantification framework for HSIC that is novel in deep learning and hyperspectral literature. We point out scenarios when UCE cannot separate ID and OOD nodes. To mitigate the limitation, we leverage inherent physical characteristics of HS data and edge-preserving regularization to propagate evidence in the spatial domain, leading to unmixing regularization (UR) and evidence- based total variation (TV), respectively. We conduct experiments on three datasets to demonstrate the effectiveness of the proposed regularizations. As the effective- ness of the UR term largely relies on the assumption of the linear mixing model (9). we will develop a more stable HSCI model subject to errors introduced by inaccurate mixing model and mixing ma- trix (please refer to Appendix G for limitations of the proposed approach). Other future directions include using superpixels to build the graph for the sake of complexity and multiple OOD material categories (as opposed only one in this work).
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ACKNOWLEDGMENTS
We would like to thank Jocelyn Chanussot at the Grenoble Institute of Technology for his helpful discussion on the hyperspectral unmixing problem and benchmark datasets. This work partially supported by the National Science Foundation (NSF) under Grant No. 2414705, 2220574, 2107449, and 1750911.
REFERENCES
Moloud Abdar, Farhad Pourpanah, Sadiq Hussain, Dana Rezazadegan, Li Liu, Mohammad Ghavamzadeh, Paul Fieguth, Xiaochun Cao, Abbas Khosravi, Rajendra Acharya, et al. A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Information Fusion, 76:243-297, 2021.
Muhammad Ahmad, Adil Mehmood Khan, and Rasheed Hussain. Graph-based spatial-spectral feature learning for hyperspectral image classification. IET image processing, 11(12)1310-1316, 2017.
Muhammad Ahmad, Sidrah Shabbir, Swalpa Kumar Roy, Danfeng Hong, Xin Wu, Jing Yao, Adil Mehmood Khan, Manuel Mazzara, Salvatore Distefano, and Jocelyn Chanussot. Hyper- spectral image ielafficatio-ttraiitiinn to deep models: A survey for future prospects. IEEE journal of selected topics in applied earth observations and remote sensing. 15:968 9999,999999999 2021.
Kartik Ahuja, Ethan Caballero, Dinghuai Zhang, -han-Christophe Gagnon-Audet, Yoshua Bengio, Ioannis Mitliagkas, and Irina Rish. Invariance principle meets information bottleneck for out-of- distribution generalization. Advances in Neural Information Processing Systems, 34:3438 3450,
2021.
Hamail Ayaz, Muhammad Ahmad, Ahmed Sohaib, Muhammad Naveed Yasir, Martha A Zaidan, Mohsin Ali, Muhammad Hussain Khan, and Zainab Saleem. Myoglobin- bbsed classification of minced meat using hyperspectral imaging. Applied sciences, 10(19):6862, 2020.
Emrecan Bati, Akın Çalıskan, Alper Koz, and A Aydin Alatan. Hyperspectral anomaly detection method based on auto-encoder. In Image and Signal Processing for Remote Sensing XXI, volume 9643, pp. 220-226 2200226666600 Spie, 2015.
Viktor Bengs, Eyke Hüllermeier, and Willem Waegeman. Pitfalls of epistemic uncertainty quantifi- cation through loss minimisation. In Advances in Neural Information Processing Systems, 2022.
Jignesh S Bhatt and Manjunath V Joshi. Deep learning in hyperspectral unmixing: review. In IGARSS 2020-2020 IEEE International Geoscience and Remote Sensing Symposium, pp. 2189- 2192. IEEE, 2020.
Marin Biloš, Bertrand Charpentier, and Stephan Günnemann. Uncertainty on asynchronous time event prediction. Advances in Neural Information Processing Systems, 32, 2019.
Bertrand Charpentier, Daniel Zügner, and Stephan Günnemann. Posterior network: Uncertainty es- timation without ood samples via density-based pseudo-counts. Advances in Neural Information Processing Systems, 33:1356-1367, 2020.
Bertrand Charpentier, Oliver Borchert, Daniel Zügner, Simon Geisler, and Stephan Günnemann. Natural posterior network: Deep bayesian predictive uncertainty for exponential family distribu- tions. In International Conference on Learning Representations, 2022.
Bohan Chen, Yifei Lou, Andrea L. Berotzzi, and Jocelyn Chanussot. Graph- active learning for nearly blind hyperspectral unmixing. IEEE Transactions Geoscience and Remote Sensing, 2023.
Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. International conference on machine learning, pp. 1725-1735. PMLR, 2020.
10

---

Published as a conference paper at ICLR 2024
Yushi Chen, Zhouhan Lin, Xing Zhao, Gang Wang, and Yanfeng Gu. Deep learning- based classifi- cation of hyperspectral data. IEEE Journal of Selected topics in applied earth observations and remote sensing, 7(6):2094-210) 2107, 2014.
Liam Collins, Hamed Hassani, Mahdi Soltanolkotabi, Aryan Mokhtari, and Sanjay Shakkottai. Provable multi-task representation learning by two-layer relu neural networks. arXiv preprint arXiv:2307.06887, 2023.
Jesse Davis and Mark Goadrich. The relationship between precision-recall and curves. In Proceedings of the 23rd international conference on Machine learning, pp. 233-240, 2006.
Christian Debes, Andreas Merentitis, Roel Heremans, Jürgen Hahn, Nikolaos Frangiadakis, Tim van Kasteren, Wenzhi Liao, Rik Bellens, Alcksandra Pizurica, Sidharta Gautama, et al. Hyperspectral and lidar data fusion: Outcome of the 2013 grss data fusion contest. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, ((6):2405-2418, 2014.
Lucas Drumetz, Miguel-Angel Veganzones, Simon Henrot, Ronald Phlypo, Jocelyn Chanussot, and Christian Jutten. Blind hyperspectral unmixing using an extended linear mixing model to address spectral variability. IEEE Trans. Image Process., 25(8):3890-399 -905, 2016. doi: 10.1 10.1109/T1..2016. 2579259.
Lucas Drumetz, Jocelyn Chanussot, and Christian Jutten. Spectral unmixing: A derivation of the extended linear mixing model from the hapke model. IEEE Geoscience and Remote Sensing Letters, 17(11:1166-1180 2019.
Ganghui Fan, Yong Ma, Xiaoguang Mei, Fan Fan, Jun Huang, and Jiayi Ma. Hyperspectral anomaly detection with robust graph autoencoders. IEEE Transactions on Geoscience and Remote Sensing, 60:1-14, 2021.
Maoyuan Feng, Wendong Chen, Yunxiu Yang. Qin Shu, Hongxin Li, and Yanyan Huang. Hy- perspectral anomaly detection based on tensor ring decomposition with factors tv regularization. IEEE Transactions on Geoscience and Remote Sensing, 2023.
Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference machine learning, pp. 1059-1059.
PMLR, 2016.
Johannes Gasteiger, Aleksandar Bojchevski, and Stephan Günnemann. Predict then propagate: Graph neural networks meet personalized pagerank. arXiv preprint arXiv:1810.0599 2018.
Renlong Hang. Qingshan Liu, Danfeng Hong, and Pedram Ghamisi. Cascaded recurrent neural networks for hyperspectral image classification. IEEE Transactions on Geoscience and Remote Sensing, 57(8):5384-5394, 2019.
Xin He, Yushi Chen, and Lingbo Huang. Toward trustworthy classifier with deep enn: Uncertainty estimation meets hyperspectral image. IEEE Transactions Geoscience and Remote Sensing, 60:1-15, 2022.
Dan Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and -ut-of-distrobution examples in neural networks. arXiv preprint 1610:22610.02133 2016.
Danfeng Hong, Naoto Yokoya, Jocelyn Chanussot, and Xiao Xiang Zhu. An augmented linear mixing model to address spectral variability for hyperspectral unmixing. IEEE Transactions on Image Processing, 28(4): 1923-1938, 2018
Danfeng Hong, Lianru Gao, Jing Yao, Bing Zhang, Antonio Plaza, and Jocelyn Chanussot. Graph convolutional networks for hyperspectral image classification. IEEE Transactions Geoscience and Remote Sensing, 59(7):5966-5978, 2020.
Xing Hu, Chun Xie, Zhe Fan, Qianqian Duan, Dawei Zhang, Linhua Jiang, Xian Wei, Danfeng Hong, Guoqiang Li, Xinhua Zeng, et al. Hyperspectral anomaly detection using deep learning: A review. Remote Sensing, 14(9):1973, 2022.
11

---

Published as a conference paper at ICLR 2024
Marian-Daniel lordache, José M Bioucas-Diis and Antonio Plaza. Total variation spatial regulariza- tion for sparse hyperspectral unmixing. IEEE Transactions on Geoscience and Remote Sensing, 50(11):4444-4555 2012.
Tao Jiang, Yunsong Li. Weiying Xie, and Qian Du. Discriminative reconstruction constrained gener- ative adversarial network for hyperspectral anomaly detection. IEEE Transactions Geoscience and Remote Sensing, 58(7):4666-4679 2020.
Audun Josang, Jin-Hee Cho, and Feng Chen. Uncertainty characteristics of subjective opinions. In 2018 21st International Conference on Information Fusion (FUSION), pp. 1998-2005. IEEE, 2018.
Audun Jsang. Subjective Logic: A formalism for reasoning under uncertainty. Springer Publishing Company, Incorporated, 2018.
Muhammad Hussain Khan, Zainab Saleem, Muhammad Ahmad, Ahmed Sohaib, Hamail Ayaz, and Manuel Mazzara. Hyperspectral imaging for color adulteration detection in red chili. Applied Sciences, 10(17):5955, 2020.
Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional net- works. In International Conference on Learning Representations, 2017.
Agustinus Kristiadi, Matthias Hein, and Philipp Hennig. Being bayesian, even just bit, fixes overconfidence in relu networks. In International conference on machine learning, pp. 5436- 5446. PMLR, 2020.
Balaji Lakshminarayanan. Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances neural information processing systems, 30, 2017.
Daniel D Lee and Sebastian Seung. Learning the parts of objects by non-negative matrix factor- ization. Nature, 401(6755) 791, 1999.
Wei Li, Guodong Wu, and Qian Du. Transferred deep learning for anomaly detection in hyperspec- tral imagery. IEEE Geoscience and Remote Sensing Letters, 14(5):597-601, 2017.
Qingshan Liu, Feng Zhou, Renlong Hang, and Xiaotong Yuan. Bidirectionalccovvolttinaaal Istm based spectral-spatte feature learning for hyperspectral image classification. Remote Sensing, (12):1330, 2017.
Xiaoqiang Lu, Wuxia Zhang, and Ju Huang. Exploiting embedding manifold of autoencoders for hyperspectral anomaly detection. IEEE Transactions on Geoscience and Remote Sensing, 58(3): 1527-1537, 2019.
Haobo Lyu and Hui Lu. Learning transferable change detection method by recurrent neural net- work. In 2016 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), pp. 5157-5160. IEEE, 2016.
Yao Ma, Xiaorui Liu, Neil Shah, and Jiliang Tang. Is homophily necessity for graph neural networks? arXiv preprint arXiv:2106. 106.006666444 2021.
Andrey Malinin, Anton Ragni, Kate Knill, and Mark Gales. Incorporating uncertainty into deep learning for spoken language assessment. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pp. 45-50, 2017.
Ekaterina Merkurjev, Justin Sunu, and Andrea L Bertozzi. Graph mbo method for multiclass seg- mentation of hyperspectral stand-off detection video. In 2014 IEEE International Conference Image Processing (ICIP), pp. 689-693. IEEE, 2014.
Debabrata Pal, Valay Bundele, Renuka Sharma, Biplab Banerjee, and Yogananda Jeppu. Few-shot open-set recognition of hyperspectral images with outlier calibration network. In Proceedings of the IEFE/CVF Winter Conference Applications Computer Vision, pp. 3801-3810 2022.
12

---

Published as a conference paper at ICLR 2024
Allan Pinkus. Approximation theory of the mlp model in neural networks. Acta numerica, 8:143- 195, 1999.
Anyong Qin, Zhaowei Shang, Jinyu Tian, Yulong Wang, Taiping Zhang, and Yuan Yan Tang. Spectral-spatial graph convolutional networks for semisupervised hyperspectral image classifi- cation. IEEE Geoscience and Remole Sensing Letters, 16(2):241-245, 2018.
Jing Qin, Harlin Lee, Jocelyn T Chi, Lucas Drumetz, Jocelyn Chanussot, Yifei Lou, and Andrea L. Bertozzi. Blind hyperspectral unmixing based on graph total variation regularization. IEEE Transactions Geoscience and Remote Sensing, 59(4):3338-3351 2020.
Zainab Saleem, Muhammad Hussain Khan, Muhammad Ahmad, Ahmed Sohaib, Hamail Ayaz, and Manuel Mazzara. Prediction of microbial spoilage and shelf-life bakery products through hyperspectral imaging. IEEE Access, 8.176986-17668 17999,, 2020.
Murat Sensoy, Lance Kaplan, and Melih Kandemir. Evidential deep learning to quantify classifica- tion uncertainty. Advances in neural information processing systems, 31, 2018.
Farideh Foroozandeh Shahraki and Saurabh Prasad. Graph convolutional neural networks for hyper- spectral data classification. In 2018 IEEE global conference on signal and information processing (GlobalSIP), pp. 968-9722 IEEE, 2018.
Maximilian Stadler, Bertrand Charpentier, Simon Geisler, Daniel Zügner, and Stephan Günnemann. Graph posterior network: Bayesian predictive uncertainty for node classification. Advances in Neural Information Processing Systems, 34:184341 18003-180444 2021.
Hongjun Su, Zhaoyue Wu, Huihui Zhang, and Qian Du. Hyperspectral anomaly detection: survey. IEEE Geoscience and Remote Sensing Magazine, 10(1):64-99) 90, 2021.
Damien Teney, Yong Lin, Seong Joon Oh, and Ehsan Abbasnejad. Id and ood performance sometimes inversely correlated on real-world datasets. arXiv preprint arXiv:2209. 00613, 2022.
Sheng Wan, Chen Gong, Ping Zhong, Shirui Pan, Guangyu Li, and Jian Yang. Hyperspectral image classification with context-aware dynamic graph convolutional network. IEEÉ Transactions Geoscience and Remote Sensing, 59(1):977-61, 2020.
Minghua Wang, Qiang Wang, Danfeng Hong, Swalpa Kumar Roy, and Jocelyn Chanussot. Learn- ing tensor low-rank representation for hyperspectral anomaly detection. IEEE Transactions on Cybernetics, 53(1):679 2022.
Qitian Wu, Yiting Chen, Chenxiao Yang, and Junchi Yan. Energy-based out-of-distributton detection for graph neural networks. arXiv preprint WwXi::3302.02944 2023.
Weiying Xie, Xin Zhang, Yunsong Li, Jie Lei, Jiaojiao Li, and Qian Du. Weakly supervised low- rank representation for hyperspectral anomaly detection. IEEE Transactions Cybernetics, 51 (8):3889-3900, 2021.
Yichu Xu, Lefei Zhang, Bo Du, and Liangpei Zhang. Hyperspectral anomaly detection based on machine learning: An overview. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 15:3351-3364 2022.
Zhuoning Yuan, Dixian Zhu, Zi-Hao Qiu, Gang Li, Xuanhui Wang. and Tianbao Yang. Libaue: A deep learning library for x-risk optimization. arXiv preprint arXiv:2306.03060 2023.
Xujiang Zhao, Feng Chen, Shu Hu, and Jin-Hee Cho. Uncertainty aware semi-supervised learning on graph data. Advances Neural Information Processing Systems, 33: 13827-72-88 2020.
Jiong Zhu, Yujun Yan, Lingxiao Zhao, Mark Heimann, Leman Akoglu, and Danai Koutra. Beyond homophily in graph neural networks: Current limitations and effective designs. Advances in neural information processing systems, 33:7793-7804, 2020.
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A PROOFS FOR THEORETICAL RESULTS
A.1 PROOFS FOR LIMITATIONS OF UCE
Suppose z E 'D TDD is point in the embedded space and y E +13 is its binary class label. An MLP-based ENN has the lower and upper bounds for the UCE loss:
Proposition 1
[e-v(2;0) 0)1
Cy 62: +1 & U) I
(18)
ey(z:
where (y(2; 0) refers the output evidence for the classs y and oz(z) e z: +1. ff0 can predict correctly: y(e+ (z; 0) e (z; 0)) > O, we have the following tighter upper bound:
UCE( y: 0) S UCE(((Z), y; 0; = r r+
(19)
ey (y( 0)
where r = 0 if the output activation function is ReLU and r 1 if it is the exponential function.
Proof. The UCE loss function has an analytical form:
UCE(a(z). y;
Ep-Di((paa(x)) |- P
(ee + e-, 0) + 2) ();
where (-(-) is the digamma function and -y refers to a different class label other than y. For example, if y +1, then -y -1 refers to the negative class. As (r()) is a monotonic increasing function, we have the lower bound:
UCE(a(Z), v:
(eee(2z 0) + e-y 0 + 2 ()) +1
(ee 0) |e- 0) 2) (z; +1
W (e, 0) + 2) (e, 0) +
It follows from the recurrence relation of the digamma function, i.e. ((x+1) = V +/x Vx >>>0 that a lower bound of the UCE loss function:
UCE(a(Z).y: 0)
(e,,(z))) + (e,, +1)
(20)
Ey c,(2;))
Similarly, we can achieve an upper bound:
UCF(a(2)) 0)
(ee 0) -e- (2; (ee (2; 0) +1)
(([e_y, (z; 0)1 + e.y (z; 2) (e 0) 1)
Te (x:01++1
Te_y (z: 0) +1
(2: 0 +i
Cy (2;
get desired upper bound in 18).
In the separable case, i.e. y(e, (2; 0) - e- (2; 0)) > O, which implies that the training examples in
the training set DD, i.e., (2, y) can be correctly classified based on the projected class probabilities:
(p= (Z) p- (&)) ) > 0, where (2). (_ (2)] [(e+ (2: 0) + 1)/S, (e_(2: 0) + 1)/S) and
e+ (2; 0) e_ (2; 0) + 2.
The last MLP layer of the ENN can be defined as:
e(z; 0) [e4 0), e (2; 0)] (o(w" 2 + b), -0-ww'p - b)],
(21)
where 2 is the input to the last MLP layer 0 {w, bw. E RDD b E R. and o(.) is the activation
function (e.g. ReLU and exponential) that outputs evidence values. If the configuration separate the example (x, y) correctly: y(e_ (2; 0) -0 (2; 0)) O. we have that c, 0) > e e2 0)
can
for y = +1 and e (2; 0) < e (2: 0) for = 1. We discuss two types of activation functions separately.
a) Exponential function: exp(w T b) > exp( w w for = +1 and exp w +b) < cxp( sxp(-wT's b) for 11, which implies that 0 S exp( hop(-wTTp 2 b) < for +1 and < exp(w" TT + b) < 1 for y = -1. It follows that: [e_, (2, 0)7 S 1. Therefore, we obtain tighter upper bound in Equation (9) with =1.
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b) ReLU function: ReLU(w b) 0 and ReLU w b) when y +1.
When y = -1, we ReLU(w"2 2 b) = 0 and ReLU(- w T b) > 0. we then have:
[e [e- 0)] O. We obtain tighter upper bound in Equation (9) with 0.
Lemma 2. Assume that the universal approximation property holds for an MLP-based ENN, i.e., ENN can learn an arbitrary mapping function from the feature vector z to the evidence values of binary classes: e(z; 0) [e+ e); 0)}² E [0, x))². Given a training set D {( (zz'yy yy)) y²) and loss function UCE(D, 0). the model optimization can get a minimal UCE loss for each training sample, i.e., 0- = arg ming UCE(z, y. 0). V(z y) D. Additionally, we have,
lim ey (z; 0) +00 UCE(ZYy,) lim e- (2; 0. UCE(2.y,0)
(22)
(23)
Proof. Thanks to the universal approximation property, the optimal solution 0* can predict the Dirichlet distribution Dir(x*) that has the minimal UCE loss for each training example (z, y) E D. Minimizing UCE(z, y, 0) over f is equivalent to minimizing over or, i.e.,
min UCE(Z, y, 0) min pe Dir(a) 8)].
(24)
where (pE y) denotes the standard cross entropy function. As tc is convex, we apply the Jensen's inequality to get,
EpeDin(a) v)l 2 lce (PE(PpEDia())))..))
(25)
Denote p = E PppDir(a) [p] = p-]. Minimizing the right side of Equation 25 we have Py - and p-y + 0 (Note, we consider that the probability never achieves zero). We also have the analytical expectation of a random variable following the Dirichlet distribution, i.c., P = a =
In a a- }. Together with the equation a = c(2; 0) + 1, we have cy 0*) - to0 and ta ta e-y(z;0*) - 0, as UCE(z, y. 0) - 0.
Theorem 1
We assume that (i) feature vectors belonging to classes {+1} follow Gaussian distri- butions with the same covariance matrix and the means =ut, respectively. i.e. P(z, y) = Py = +Z)N(z; μt, Σ) + PPy = -1) - μ. E). with Fy +1) = Fa Py -1) = 0.5: (ii) the optimal solutions O that minimize (e (z)
( 0) can separate both classes: e, 0) > e-y(z; 0), V(z, v). Let a() be the exponential function. The optimal solution 0 (w+,br is the same as the optimal solution of LDA, i.e.,
w Σ and b² = 0.
(26)
Proof. According to the assumption on Gaussian distributions for the two classes, the upper bound of UCE loss, i.e., UCE, has the following relations:
P(z,))P(a.)) yC)
1 Ban UCB(((()) +1;0)= - = D2,
1
2
2
E c4(2;0) Ez-N(-n.E)) (z;
Ex~N(H exp( -w" T - b) +
exp(w" z z b).
(27)
It holds- -w - b ~ N(ww" b, w Ew) for z ~ N(H, Σ) and w z + ~ N(-wTTu 4
b, w T Ew) for ~ NV u. E). Using Elexp(r)) = exp(p + 0² /2/22 for r ~ N V1,,0* 02), we get
Ex E(,,(( [exp(- ww""" z b)] -exp(-w" 4 b+ w" (2//2)
(28)
Eg [exp(w? 2 + b)] = =exp(-w's / + b w Ew/2)
(29)
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Plugging Equations 28
and 29
into Equation 27) yields
E ((x.))~((z..) UCE(a
exp( pr w Ew 22) + exp( ( w +b+ w E'Eww' Ew/2)
= exp(- w p + w Σw (exppp())pp))))) + exp(b)).
Therefore, the minimization problem of UCE can expressed
min (e..)) UCE(a(Z),
minexp( w μ + w Ew Ew/2) (xwpp(exxp(pec exp(b))
w.b
min exp( w μ + w Σw ) min(exp( + exp(b))
It is straightforward to obtain the optimal solution for b, that is,
b" = arg min(exp(-b) + exp(b)).
(30)
As the exponential function is monotonic, we have an equivalent minimization for follows
min exp( w
EwwwEww/2 22)
min ww" r + w Ew/2.
(31)
By taking the gradient of equation 31 with respect to w and setting it zero, we obtain - Σw O, which has a closed-form solution, given by
w E
(32)
A.2 GRADIENT ANALYSIS OF PROPOSED UNMIXING-BAN -AASED REGULARIZATION TERM)
Proposition 2, Assume the linear model O holds without noise, the gradient descent for minimizing the UR regularization increases the predicted evidence of ground-truth class for ID instances and decrease the total evidence for OOD instances with the corresponding pure material contained in the pixel; formally, we have
a) For an instance (ar' ( yy) with feature matrix x² .my and yy E ((I....,CO C. the ground truth ID class label, one has
aUR(E) de < 0.
(33)
g¹
b) For an OOD instance instance (x', ..) with x' = m and y² = 0 f1....,CH
aUR(e) del 2 0.
(34)
Proof. Given an instance (x )²) with x E RD, y E (1....,C, o.... ID material signatures m. E R for {1...., C). the OOD material signature m E R optimized in Appendix [A.3] and the subjective logic opinion wt = (b', u²) is based on model prediction e())). For brevity, we omit 0 in the rest of the proof, thus getting b' formulated as
C The UR term can be
c+ of
C+E
DC
e.mo
Cm
UR(e) = Eiey 1|
C+
Ec=ret
el
C+
Taking the partial derivative ef, which is the evidence scalar of class for instance i, we obtain
aUR(e) Oel k
EC e,ma + Cm,
T (Eeyk et + C)m= Enyk e.m. Cm
-2 x²
C+ CC DE-ee eg
(C + DcTed
(35)
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If y' E fl..... .} and æ' my which indicates the instance is a pure ID material, then have the partial gradient with respect to the ground truth class y as follows:
(IC Ecerc emm, Eosyyt m. + Cmm)) (Conpu ef Cym (Erogye e.mo Cmm))
(UR(e) De,
(C > og)a
20)
|10 Exeer m) CEery, e.m, + Cmb)lla (C ΣE og)²
50.
(36)
If y' = and x' =m,, which implies pure OOD material, then it further follows from 35 that
(10 Eo Jm - (Eous e.m - Cm.)) (EEnen e + CJm (Eooen m.m. + Cm.)
OUR(E) Oct
(C 2 egys
-2 -E m E.)) (C +c= 0.)
Σ c +0mm
E De +mm))
(37)
The gradient descent to update the total evidence can be expressed as
(UR(e)
C
C
aUR(e)
c :=
and
E
c
(38)
dek
dek
where 8 is the learning rate and the summation of class-wise gradient is calculated by
-2 22 e.m; - Eee e.m.)"
(UR(e)
c +
Clmy
Σ
cmm
Cm
Dec
ΣE E.=1e))) 2,)
-2 (IC c. - Ee e.mo)
Σ
m: 0 me Σ
Σ
(C
1e Cmm,
-._ cc)²
ID cmm;
Ee cmme)
+ Ee e,)a
me
m
Σ c (C 1)
Without loss of generality, we assume that there is only one ID class, ie. C 1. then we have
aUR(e)
(E Re)
Imm - mID l 2
dej
(C + D ec)
A.3 ANALYTICAL SOLUTION FOR SIGNATURE OF OOD MATERIAL
Given the feature set of the whole graph {a E V) and each instance has an associated evidence vector e¹ (0) by some fixed 0, the optimal OOD material's signature by minimizing UR UR(m.) over the graph has an analytical form as
Eiev(æ' E.e.m St
m.
(39)
Eiev gia
Proof. The summation of UR term over the graph can be expressed by
En em
C
UR(m.)) = Σ ||
m,
12
(40)
St.
S
Taking the derivative of 40 with respect to mon we have
aUR(m.) 2
Ee
c.m.
C
C
(x)
mm))l
5
3m,
S
(41)
=-20
Σ c
Σ.
e
St
St
1
Σ
Siz
ma
Lie
iev
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Setting
mm) 8m. 0 leads to
m. Eiev(a' E m. 5
(42)
Σiev g
B DATASET DETAILS
HSI captures data at various wavelengths for given spatial region. Unlike the human eye, which possesses only three color receptors sensitive blue, green, and red light, HSI precisely measures a large range of the light spectrum for every pixel in the scene acquired by sensors such as Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor. It offers detailed wavelength resolution not just within the visible range but also in the near-infrared range.
We use three widely used HSI datasets in the HSIC task: the University of Pavia (UP), the University of Houston (UH), and the Kennedy Space Center (KSC). We present details of these three datasets in Table 3 Note that all three datasets used here show high homophily scores Zhuu al. (2020) (exceeding 79% is regarded
high homophily score), indicating that pixels with similar spectral features tend to belong to the same category.
Table 3: Summary of the three HSI Datasets used for experimental evaluation: "Spatial" the 2D dimension in terms of width and height; "Spectral" is the spectral bands, i.c. the number of features for each pixel within the "Wavelength" range (nm): "Labeled pixels" counts the labeled pixels in the ground truth datasets and we do not care about the unlabeled pixels; "Training ratio" calculates the proportion of training pixels across all labeled pixels; and the "homophily score" measures how likely nodes with the same label are near each other in graph.
Spatial
610
610
340 1905
512 614
Spectral
103
144
176
Wavelength
430-860
0.35-1.05
400-2500
Labeled pixels
42,776
17,270
4,364
Categories
15
13
Training ratio
8.67%
27%
7.72%
Homophily score
The UP dataset was acquired by Reflective Optics System Imaging Spectrometer (ROSIS) sensor during flight campaign over the university campus at Pavia, Northern Italy. The detailed class description and training/validationn
ratio are presented in Table 4 following the same setting
as (Hong et al. 2020
Table 4: Land-cover classes of the UP dataset.
Training
Validation
Asphait
327
1260
5044
Meadows
503
3629
14517
Gravel
284
363
1452
Trocs
152
582
2330
Painted sheets
Bare Soil
457
914
3658
Biturnen
349
196
785
Sell Blocking Bricks
318
672
2692
Shadows
152
159
636
The UH dataset is collected by the Compact Airborne Spectrographic Imager (CASI) and released data fusion contest by The IEEE Geoscience and Remote Sensing Society. Table 5 presents the classes and dataset split following the contest
com
TGRS GCN
201 daaa i
commi wtt.iiteteeeeeeeee hts.//aaaaaffuiiioo fusion/
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Table Land-cover classes the UH dataset.
Class No.
Class Name
Training
Validation
Test
Healthy grass
198
235
Stressed grass
190
252
1012
Artilicial Lurf
227
113
455
Evergreen
188
215
861
Deciduous trees
186
222
890
Water
196
256
1024
Residential baildings
191
232
931
Non-residential buildings
193
272
1089
Roads
191
246
987
Sidewalks
234
266
1066
Crosswalks
192
247
900
Major thoroughfares
Highways
213
60
240
Railways
227
114
457
Toral
3068
2835
T1367
The KSC dataset was gathered by Airborne Visible/Infrared Imaging Spectrometer (AVIRIS). There no widely-used public split for KSC and we randomly pick 20 nodes from each class for training following (Kipf & Welling 2017). The detailed class description and training/valicn training/validationnees ratio are presented T
Table 6: Land-cover classes of the KSC dataset.
Class Name
Training
Test
Scrub
20
111
504
Willow swamp
20
152
Cabbage palm hammock
20
160
3
Cabbage palm/oak hammock
20
34
158
Slash pine
S
Oak/breddead hammock
20
31
142
Hardwood swamp
20
58
Gramioodd marsh
20
280
8
Spartina marsh
20
75
340
Cartail
20
261
10
Salt marsh
20
59
272
Mud
12
Wate
20
136
616
Total
260
737
3367
C GRAPH CONSTRUCTION
Considering most HSI datasets contain a limited number of labeled pixels, we follow recent work Hong al. (220)) to build the graph only using these labeled pixels while ignoring the pix- els that do not have ground-truth labels. We model the labeled pixels in one HSI scene as graph (V, E), where V (1,2,... N} denotes vertex and E. C V V is the edge set. Edges can be represented as a weighted adjacency matrix W E R. NxN and the weight is calculated with the radial basis function the similarity between node features Qin al. (2020).
W5i exp
d(æs x> )/
where d(x',x') ) is the distance between two vertices i and j. such as the Euclidean distance or cosine similarity, and a control parameter for the similarity. We use the cosine similarity
<
d(xx; x
x
x
thanks to its scale invariance, based the observation that illumination alters the scaling of spectra while preserving their overall shape in the spectral domain (Merkurjev et al. 2014). To improve the computation efficiency with better scalability, only keep the first K largest weights, considered as nearest neighbor, for each node to build a sparse graph. We choose experiments.
and = 0.1 in the
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The built graph exhibits strong homophily characteristic where nodes typically associate with others that are "similar" or "comparable" from the perspective their respective categories. It has been shown that GCN manages such highly homophilic graphs effectively (Ma et al. 2021.
D MODEL DETAILS
For comparative analysis, employ five bascline methods. First is classification model that utilizes entropy as its uncertainty metric. Additionally, we select three representative anomaly detection techniques, mentioned in recent review (Xu al. (002).. Furthermore, we incorporate two state-of-the-an uncertainty quantification models for semi-supervised node classification.
Pseudocode for model
We provide the pseudo-code for the OOD detection in Algorithm
Algorithm Model Optimization for OOD Detection
1: Initialize neural network parameters 0" and OOD endmember ml randomly and set k:
2: repeat
3: Fix m then optimize neural network parameters using regularized learning function (Equa-
tion 17 and get the 0
4: Get the prediction of beliefs and epistemic uncertainty (i.e abundance coefficient) with @
5: Fix 0++1, calculate the optimal m based on Equation 12
6:
A ++
7:
until convergence
8: Output optimized 0" = 4 and m = m
For GCN-based models, we use two graph convolution layers and 0.5 dropout probability. Following the graph size, KSC, UP, and UH have hidden dimensions of 64, 128, 256, respectively. We use early stopping with the patience of 30, a maximum of 5,000 epochs, and validation cross-entropy as stop metric. For all models, we use the Adam optimizer, and the learning rate and weight decay are carefully tuned for each dataset.
Seftmax-GCNi We classic two-layer GCNs optimized with cross-entropy loss (Hendrycks) & Gimpel 2016) based on the assumption that correctly classified examples tend to have greater maximum softmax probabilities than erroneously classified OOD examples.
Table 7: Hyperparamters for noftmax-GCN Model
dittsset
UP
LO0E-02
L5
UH
1.005-02
1.008005
TLRSR.
Tensor Low-Rank and Sparse Representation (TLRSR) model (Wang et al. 2022) used principal component analysis (PCA) method bands, followed by a tensor low-rank framework to preserve the inherent HSI structure, for ex- tracting the LR background part the dictionary of TLRSR. Following their parameter tuning, search A and A2 from the set {0.001, 0.005, 0.01, 0.05, 0.1, 0.2, 0.33. The hyperparameters used in our experiments are listed in Table
preprocessing step to exact subset of HSI
RGAE. Robust Graph AutoEncoder (RGAE) (Fan et al. 2021 is a modified autoencoder framework combined with gradient normalization of each sample to make more robust to noise and anomalies. Besides, it has a graph regularization term for preserving the lo- cal geometric structure of the given high-dimensional data. Three hyperparameters need to be tuned carefully, i.e., (1) Trade-off parameter X that balances the regularization term and the range set to {10-4, 10-3 10 10-2.100'): (2) Number
superpixels and the rage is (50, 100, 150. 300, 500]; (3) Dimension of hidden layers Mhid and the range is
(20, 40, 60, 80, 100, 120, 140, 160). We use the validation OOD ROC to pick the top performance
models. The hyperparameters used in our experiments are in Table
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TRDETVAD. Tensor ring (TR) decomposition with total variation (TV) regularization model (TRDETVAD) (eng et al., 2023) is proposed for hyperspectral anomaly detection. This method decomposes hyperspectral imagery data into background and anomaly components, leveraging the low-rank nature of the background across spatial and spectral dimensions, while employing the TV regularization to enhance the piecewise smoothness of the background. Two tuning parameters as the weight of TV regularization and 3 as the weight of sparse regularization term. These both searched from {0.01, 0.05, 0.1, 0.5, 1, 5,10). We the OOI) AUROC value on the validation set to select the best performance models. The hyperparameters used in our experiments are in Table
two parameters
Table 8: Hyperparamters for anomaly detection baselines
Model
parameters
hyper
4
6
UP
7
8
UH
2
10
5
KSC
RGAE
300
100
100
100
150
150
150
500
150
160
120
120
120
80
80
20
40
160
150
40
40
Ag
0.01
0.01
0.01
0.01
0.001
0.001
0.005
0.05
0.001
0.01
TRDFTVAD
0.31
0.05
0.01
0.01
0.05
10
8.a.
0.01
0.08
0.05
10
0.01
10
EGCN-GKDE. Similar to ENN, EGCN uses activation layer instead of softmax output non- negative values as the parameters for the predicted Dirichlet distribution. The representation learning step uses GCN layers, while GKDE associated with EGCN is designed to estimate prior Dirichlet distribution parameters for each node, which is calculated based on the shortest path between test nodes and training nodes belonging different classes the graph. A necessary condition is that nodes with a high epistemic uncertainty are far away from training nodes and nodes with a high aleatoric uncertainty are near the boundary of classes. It may not exhibit good performance when this condition is not satisfied, i.c., OOD is close to ID training nodes. We present the detailed analysis of GKDE in Section
In our experiments. we also integrate the GKDE teacher by default. we use 3 0.5 for KSC and UP, 3 = 0.2 for UH. For hyperparameter tuning, we first tune the learning rate (Ir) and weight decay (wd), as well as the trade-off parameter for GKDE teacher in X2 based on the average result of II accuracy and 00DDMisclasssficatinn ROC. We suppose there are OOD classes involved in the validation set for hyperparameter selection for all models. The hyperparameters used in experiments are presented Table9and and Table 10 for misclassification detection and OOD detection, respectively.
Table Hyperparamters for Misclassification Models
datteset
wd
GKDE A1
Aa
Ag
wd
GPN
A1
Ag
Ag
1.00F-022
1.00F-44
1,005-04
1.00F-03
3.00F-03
1.00F-03
1.00F-022
1,00F-002
UII
1.00--02
1.00E-04
1.00-022
1.00-04
1.00E-05
1.00E-04
1.00E-04
1.00E-04
1.00E-01
1.00E-04
KSC
1.00-022
1.00-004
1.00-003
1.000-02
1.00F004
1.000-02
1.00F-03
1.00F-03
1.00F-04
1.00F-02
GPN. Graph posterior network (GPN) Stadler et al., 2021 applies multi-layer perceptions for representation learning, followed by normalizing flow for density estimation in the latent space. The graph structure is leveraged for evidence propagation. In detail, GPN consists of three components. (1) A feature encoder g6 maps the original node feature x R E V onto a low-dimensional latent space z' E R with simple two-layer multi-layer perception (MLP) encoder, H is the latent dimension. i.e. z² = go (x*) and Φ is the encoder parameters. (2) A Radial normalizing flow hep estimates the density of the latent space per class, which used to compute the pseudo evidence (class counts) 3 h='))) N₂ PPz'|c; sp) (3) A personalized page rank message passing scheme diffuses the pseudo counts (density multiplied by the number of training nodes) by taking the graph structures into account, i.e. a = Evev II.,, &e with II.,, is the dense PPR score reflecting the importance of node v i. Following the GPN paper (Stadler al. 2021).
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set latent space to have 10 dimensions and 10 radial layers for normalizing flow. We use the CE loss to pre-train the flow layer, teleport equal to 0.2 and propagation is iterated with 10 steps. For the parameters in the optimizer and tradeoff's for the loss function, the tuning process is the same as EGCN. The parameters used in our experiments are presented in Table9 and Table 10 for misclassification detection and OOD detection, respectively.
Table 10: Hyperparamters for OOD detedction Models
Dutinet
Typer parameters
Ir
wd
GPPN
X1
A2
A3
wd
GKDF
>g
Ag
1000-04
500E-03
100E-04
1000-00
100E-03
10DE-02
L00E-01
L00E-04
LD0E-05
UP
1. 1.000-03
1,000 03
5.00E
1.00-00
1O0G
1.00-03
100E+00
100E-0L
1.00E:01
1.00E-05
1.00E-00 02
100E-03
1000-05
100E-05
0.00E+00
0.00++00
LOOE-01
LO0E-02
LO0E 05
LO00-05
1,000
1408-04
1,08:005
1008-01
1.00.05
1000-03
L00E-02
LOOE:
LOOL 05
5006-01
1006-033
1000:05
1,00--33
LI0E-04
U11
1.008403
1008-08
1.08-03
1 1006-08
1000:04
1006-02
1006-03
L00E-02
L00E-04
LO0E-05
1 000--04
100F44
1.00.04
1 1000+00
1000-00
10F-002
100F-05
1.00-02
1. LO0FF044
100F-04
1.000-03
1,005-03
1O0E
5000--0
1008:004
1000-04
1000E403
1000400
100ES0S
LIDE-O
LIRE
L.DE
KSC
1,00E
1.00E 03
5.00E 03
5000E-00
1.00 04
1,005-00 04
L000+000
L00E+00
1.00E-05
1.00E-03
1.00E-03
1.00E-03
1.00E-05
1000E-05
LOOE 02
1.00-02
L,OGE
LOGE: 01
L.00: 02
LO0E
5,005-03
1000-08
1,00:00
Complexity analysis We provide the complexity analysis in Table 11 The three anomaly detec- tion methods (RGAE, TLRSR, TRDFTVAD) are implemented with Matlab and run on desktop with Intel Core 17-9700 and 16GB memory. The remaining three (softmax-GCN, EGCN- GPN-based) are implemented with PyTorch and tested on single GPU RTX4090 located on server with AMD Ryzen Threadripper PRO 5955WX and 256GB memory.
Table 11: Complexity Analysis
Runing Time (sccond)
Number of parameters
UP
UH
KSC
UP
UH
KSC
RGAE
1,660
11, 918
3, 899
n.a
n.a
n.a
TLRSR
40
89
91
n.a
n.a
n.a
TRDFTVAD
2,100
n.a.
10,740
n.a
n1.
n.a
softmax-Gol
177
30
126
14.9k
41.8k
24.9k
EGCN-based
320
43
129
14.8k
41.5k
24.8k
GPN-based
100
80
77
16.4k
42.9k
26.7k
n.a means either metric is not applicable model is not tested due the the time limit
E ADDITIONAL RESULTS
E. ABLATION STUDY
We conduct an ablation study in Table order demonstrate the influence of each regularization term applied to EGCN and GPN for OOD detection. The comparison includes the original regular- ization term used EGCN/GPN
well as our proposed UR and TV terms, namely R. UR, TV respectively in Equation 17 The EGCN framework inherently incorporates GKDE teacher (de- noted as AT' in T2) and leverages node-level distance metrics for OOD discernment. The GPN framework applies entropy regularization (denoted as ER in the table) that encourages non-peak prediction. We use the UP dataset for the ablation study. Given the results presented in Table [3 and Table 14 show near-ideal performance for scenarios 'UP-4' and 'UP-8', consider 'Up-6" and 'Up-7" for ablation study, where classes 6 and 7, respectively, are chosen as OOD, with the remainder being treated as in-distribution (ID).
The key findings obtained from the experiment are as follows: (1) GKDE (AT) regularization yields some improvements UCE-only for 'UP-6' and shows comparable results for 'UP-7'. It indicates
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that the GKDE teacher's prior knowledge may not be always true, i.e. OOD test nodes are far away in terms of graph-based distance from the training (ID) nodes compared to ID test nodes (2) The proposed UR term significantly improves the accuracy of OOD detection compared to other baseline models. Specifically, in the context of UP-6, the UR term augments ROC by 15.97% and 50.79%, and PR by 6.44% and 8.72% for EGCN and GPN respectively. For UP-7, improvements of 6.15% and 16% in ROC together with 4.7% and 16.03% in PR are observed for EGCN and GPN, respectively. (3) The TV regularization term also enhances OOD detection in overall, with improvements ranging from 1.25% to 21.68% in terms of ROC and 1.01% to 4.18% in terms of PR. There is a slight decrease within 1% on EGCN for 'UP-7' and it may be due to reasonable randomness. (3) Models employing both TV and UR terms generally deliver the best results, except for the EGCN model on UP-6. Although the EGCN-TV-UR performs slightly lower (within I 1%) ROC and PR than the EGCN-UR, the former achieves better ID OA with 3.05% compared to the latter.
Table 12: Ablation study of regularization terms
UP 6
UP-7
OA
ROC
ROC
PR
UCE-only
77.16:00.2
12.6011110
4.0
75.0543.08
80.70=E187
21,73
Baseline
UCE-AT (EGCN)
74.86++2.37
14.82:44.45
4.50=00.77
75.91+22.25
80.51+22.13
21.72+3.56
EGCN
EGCN-UR
74.32:11.19
90.79110:22
10:4410.40
12.31
86.66.11.71
Qurs
EGCN-TV
77.1342.15
83.96.10.74
6.63-10.28
1..50-11.24
79.81-1:2.98
20.88-14.06
EGCN-TV-UR
17.37:41.78
89.03-t 0200
9.33+0.74
75.05+3.11
86.78:1:45
26.01
UCE
66.92.1.395
30.60:17.66 7.65
1.69
64.3611..82
71.58:8.50
14.16:13.94
Baseline
UCE.ER ER (GPN)
66.11117.32
39.40
2.06:10.63
64.94.1..72
14.58.112110
16.51
GPN
GPN-UR
54.67+2,44
90.19:10.46
10.78:10.34
58.7343.71
90.58:11997
32.54:1:222
Ours
TV
66.50.1 1.56
14..53
3.071
61.00.13.31
82.33.11.58
19.52.10.98
GPN-IV-UR
14.73::1.39
90.39:1:.34
11.31::0.37
66.081:4.39
92.89-10.17
42.47::0.92
Bold numhers indicate the best model across all the uncertainty metrics for each dataset.
E.2 ANALYSIS GKDE TEACHER FOR OOD DETECTION
GKDE assumes that OOD test nodes are far away in terms of graph-based distance from the training (ID) nodes compared to ID test nodes. This assumption is not always true in practice. We provide the GKDE form as follows:
fo y² #c
he dıy)
(gdd)) 3² =c
L with g(dig) exp(- 232) and di denoting the graph distance. The evidence prior é, EieL h(y', diy) with hhy', d.y) [h...... he...., hc]. B is the bandwidth in the Gaussian kernel function.
We investigate the influence of parameter 3 in the GKDE teacher on the OOD detection perfor- mance. On specific OOD setting, pixels belonging to "shadows" in the UP dataset are considered as OOD. Figure 4 shows the OOD detection performance for GKDE prior model with different 3. With 3 1, GKDE can achieve 96.40% ROC and 73.13% PR value. B 0.1 is random ranking result while 3 = 10 is much worse result.
Figure 5 presents the predicted vacuity maps obtained by a variety of 3 values, in comparison to the ground truth map where OOD is labeled 1, and ID is O. Note that unlabeled data in the hyperspectral image is marked as 0 for all sub-figures (including both ground truth and prediction figures). The choice of 3 can significantly influence OOD detection performance. Within a single sub-figure, a comparative analysis reveals a clearer distinction between OOD and regions when = 1. For 3 = 0.1, nearly all pixels exhibit high vacuity, whereas for 3 10, they display low vacuity uniformly. Hence, adjusting 3 is crucial for accessing a good GKDE prior.
In comparison to the GKDE model, which achieves 99. 99.1% % on ROC and 96.9% on PR, the GKDE prior does not perform as effectively as EGCN. As seen in Table 13 through 18 the performance disparity between GKDE and EGCN that the GKDE teachers may not yield optimal results. Conversely, in certain situations, GKDE can indeed enhance the learning process of EGCN.
reach up to 27% on ROC and 19% PR. This suggests
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10.0
Figure 4: AUROC and AUPR curves for different / of GKDE teacher in OOD detection.
Ground truth
0.1
B 0.2
0.5
WHN
04
26
5.5
320
302
320
500
103
2:00
3.00
200
300
102
230
8
10
6003
H
HHH
WHH
605
8000
0/
00
626
322
6.033
8.093
332
500
21
520
302
8052
5.003
092
2.30
1.0 200
310
192
2.50
Figure 5: Predicted vacuity maps using different 8 values in the GKDE teacher for OOD detection. The yellow pixels marked the ground truth subfigure the OOD pixels and the model is expected to predict high uncertainty for those pixels than the remaining pixels in the picture. B = has the best ability distinguish OOD pixels from ID pixels.
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E.3 PARAMETER SENSITIVITY ANALYSIS
Besides the general neural network parameters like learning rate and weight decay, our model in- corporates three specific hyperparameters denoted as λ1. A2, A3 as presented in Equation 17 The parameter Å1 is a trade-off weight for the uncertainty quantification framework, i.e. GKDE teacher for EGCN or entropy regularization for GPN. The other two hyperparameters A and X correspond to the weights for the proposed UR and TV terms, respectively. We conduct sensitivity analysis on these hyperparameters using the variations of A2 A3 and their impact on the ROC performance. Figure that the GPN model is less sensitive these parameters compared to EGCN. For in- stance, in the case of 'UP-8', the ROC metric shows stability to variations in X2 when X lies within the range of [1e-5, le-2]. Similarly for 'Up-7', the ROC metric remains consistent with changes in 22 when A3 is between [0.1, 1]. When the TV term's weight is set small, such as le-5, the model becomes highly sensitive to the UR term's weight. Within the EGCN framework, the model's per- formance is nearly unaffected by A when 22 ranges from [le-5, lc-4] for 'Up-8', but it decreases significantly when 22 exceeds le-4.
(a) UP-8-EGCN
(b) UP-S-GPN
(c) UP-NNE
(d) UP-7-GPN
Figure Sensitivity analysis of parameters (A2, As) on AUROC for OOD detection.
E.4 RATIONALS OF THE PROPOSED REGULARIZATIONS
Explanation UR performance Equation I00 assumes that the feature vector corresponding to an ID node is linear combination of ID endmembers; and consequently, if feature vector is orthonormal to the space that is spanned by M, then it is indicative of an OOD node. (1O) holds particularly well in scenarios where the OOD spectral signature is orthogonal to M. If the OOD spectral nearly parallel to one of the ID endmembers, then the UR term can not distinguish the OOD from this ID material.
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We take the 'UP-6' and 'KSC-5' two illustrative examples. From Table 13 and note that the UR term yields an improvement of 18% and 40.4% in ROC for 'UP-6', whereas the improvement for 'KSC-5' is relatively modest, -0.2% and 4.4% for the EGCN and GPN frameworks respectively. The difference in UR's improvement lies the difficulty of identifying the OOD class. We present the distribution of cosine distances between OOD nodes and IDD nodes of UP-6 and KSC-5 in Figure 7 For "UP-6', class 'Bituman' (class 6) displays relatively low degree of similarity (approximately 0.8) compared nearly 0.9 in KSC-5. This verifies that the UR-based regularization modeled in Equation (TO) can improve OOD detection performance under the assumption that OOD spectral should be as orthogonal to the ID feature space as possible.
We also emphasize that the distinctiveness of raw features between ID and OOD data does not in- herently ensure the successful detection OOD nodes when employing the UCE loss. Elaborations on this subject are provided in Section 3.1. To summarize, the UCE loss formulation only involves ID nodes, while disregarding the OOD nodes the latent space during the feature learning step. Consequently, features that are characteristic of OOD may be overlooked due the model's 's focus on ID feature separation. Purthermore, despite potential separability in the latent space, the deter- ministic boundary delineated by EGCN erroneously allocate OOD instances ID regions in the final layer of dimension reduction. Therefore, the property that UR term designed for physical relation between raw features partially mitigate some drawbacks of the UCE loss.
the
(a)
(b)
Figure 7: The distribution cosine similarity of the OOD material with all ID endmembers. Each color represents a distinct ID material. The x-axis quantifies the cosine similarity, where smaller values imply less correlation of ID and OOD, the cosine similarity is closer to suggests a high degree of similarity, which indicates difficulty in separating ID) and OOD.
Explanation of TV performance Comparing 'UH22' and 'UH-10' in Table 14, the TV brings significant improvements on 'UH-2'. This is because the TV term is trying to smooth the predicted total evidence (inverse of epistemic uncertainty) across the spatial neighbors and it may bring more gain when the spatial neighbor tends to have similar ground truth evidence. In Figure class 2 'artificial turf") more concentrated than the class 10 ("sidewalks").
When pixels of particular class are not densely concentrated, the inclusion of the TV term may not yield significant improvements nut-of-diitribution (OOD) detection but rather produce com- parable results. There may be several reasons why the combination of TV and UR terms leads to slightly lower OOD detection performance than the UR term alone. For example, there often exists trade-off between OOD detection performance and in-distribution (ID) classification performance, as discussed in a recent work (Teney et al. 2022). In the case of 'EGCN-UR-TE 'Houston-1' dataset, observe lower ROC but higher ID Overall Accuracy (OA). In Table 15, 'KSC-5' demonstrates that 'GPN-UR-TV has a lower ROC but a higher PR value. Examining the combination of these metrics provides balanced scenarios. For 'UP-6,' all three metrics are lower for "EGCN-UR-TV' This discrepancy could arise from model randomness, hyperparameter variations, and the performance differences is within a reasonable range.
applied to the
comprehensive perspective, especially for im-
Performance difference between EGCN and GPN We would like to highlight the distinctions between the two frameworks, EGCN and GPN, they may exhibit divergent performance patterns.
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(b)
Figure 8: Ground truth pixels belonging to class in (a) and class 10 in (b). The pixels are more clustered and concentrated in class 2 compared to class 10, and as result, TV is more effective (with more improvements) in setting aside class 2 as an OOD class, compared to class 10 as OOD, shown in Table
Despite both frameworks utilizing the same UCE loss function, EGCN employs discriminative boundary for prediction, whereas GPN employs a generative model for density estimation. Conse- quently, these frameworks encounter distinct challenges during the training phase.
E.5 DETAILED RESULT
For each dataset, we create four random configurations and select one class as OOD. In the main paper, we display the weighted average, factoring the count of test OOD nodes for every dataset. In this section, we show all the results for the twelve configurations in total.
Tables 13 14 display the results for UP. is worth noting that some classes are easier to classify while some are challenging. For example, class-4 is easily discernible by both anomaly detection and uncertainty quantification methods. However, our introduced framework significantly outperforms the anomaly detection benchmark. This may indicate that class-4 possesses distinct features compared to other pixels the image. On the other hand, for class-6, the PR values are suboptimal across all models, indicating that considerable number ID pixels have elevated predicted vacuities. When comparing class-7 and class-8 as the OOD class, models based GPN fare better with class-7, while those based on EGCN excel with class-8. A similar trend is observed with the GKDE teacher, whucg performs commendably with class-8 but falls short with class-7.
softmax-GCN struggles in this regard, whereas
Figure 9 presents the predicted vacuity map for different models when "shadows" (class-8) as the OOD class in UP. TLRSR and RGAE can identify the OOD all. EGCN tends predict higher vacuity scores for OOD nodes (exceeding 0.8) while GKDE has a much lower vacuity score for all nodes (below 0.001). After applying the UR term, there seem be fewer false positives and lower vacuity scores for ID nodes after applying the TV term.
We have a similar conclusion for the other two datasets. Tables 15 16 present the result for the UH dataset, while Tables [8 show the result for KSC. Take the KSC dataset for an example. We ob- serve significant improvements in OOD detection performance for our randomly picked four OOD settings from Table 15 and Table 16 (We ignore the KSC-12 scenario due to the perfect OOD detec- tion performance). Specifically, the inclusion of UR led to modest improvement of 0.3% 2.2% on ROC and up to 2.5% PR for the EGCN model, while the GPN model exhibited remarkable improvements of 4.4%-60. % on ROC and up to 9.2% on PR. Furthermore, the incorporation of TV resulted in substantial enhancements for both models. The EGCN-UR-TV model demonstrated a notable increase of up to 22.4% on ROC and 5% on PR. The GPN-UR-TV model exhibited a substantial boost of up to 3.7% ROC and 12.2% PR. Our findings indicate that the inclusion
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HHH
102
205
64
102
25
15
EGCN
08
660
CPN-UR-TV
2.000
300
380
Figure 9: Predicted vacuity map for various competing methods.
of UR and TV terms consistently enhances OOD detection performance the KSC dataset under the specified experimental settings.
E.6 ARCHITECTURE ANALYSIS
To investigate the impact of different encoder backbones on the OOD detection, we apply a re- cent deep GNN architecture (Chen et al. 2020) as the backbone for the EGCN framework, termed EGCN2. The results, as shown in Table 19 align with those obtained using GCN as the encoder, where the UR term notably enhances the ROC and PR metrics for the OOD detection. While the TV term yields comparable results of the OOD detcction, it demonstrates improved ID classifica- tion performance. Furthermore, comparison between Tables [2 and Table 199 reveals that GCN2 exhibits superior OOD detection the 'UP77' case but shows reduced effectiveness 'UP-6'.
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Table 13: OOD Detection Result for UP
UP 4
UP 6
ID
OOD ROC
OOD PR
ID OA
OOD ROC
00D PR
sollmax GCN
7..111.8
62.9114.6
6.516.1
77.412.0
29.812.77
1910.1
GKDE
a
99.6t...a
96.8+n.
71.0=aa
59.8t..
3.140.a.
RGÁE
D.a.t..a..
99.3:n...
86.22n..
n.a.t.....
60.6mna.
29:na.
TLRSR
t.a.
98..
71.4na.
n.a.
70.9+n..
3840a.
TRDFTVAD
b.a.ta...
98.34: 0.1
61.34:1 2.3
n.a.o..
52.94:244
2.340.11
EGCN
15.3=19
99.9
97.820.3
17.0224
02
4.1+0.11
EGCN UR
1.2
99.9+10.0
97.14:0.8
76.3
90.34:0.2
10.4:0.1
EGCN - UR-TV -TV
76.6#3.1
100.0=0.0
99.8=+0.00
7..8:2..
90.1+:022
10.3+0.2
71.5+-0.0
99.6-10.00
99.1-1..00
29.5
40.411554
2.140:88
GPN- UR
71.=0.11
99.6+0000
99.1+=0.0
10.9+1.7
1.
10.5+11
GPN -UR -TV
63.1=0.6
99.6+0.00
98.940.11
51.7=2..0
90.7+:0.6
11.3+008
Bold numbers indicate the best model across all the uncertainty metries for each datasct. Undertined numbers are the best results within the same model type for each dataset, n.a means either model or metric not applicable. "UP-44" takes class 4 as the OOD class and 'LP-6' takes class the ODD class.
Table 14: OOD Detection Result for UP (cont.)
UP OOD ROC
UP- 8
datasct
ID OA
0OD PR
ID OA 75.3 1.5
OOD ROC
OOD PR
softmax-GCN GCN
76.6
3..2=39
27.44.0
14.2.55.5
1.3:2..5
GKDE
a0.7
$7.0=ta..
9.8....
68.7
95.3
72.2:ta..
RGAF
n.a.
20.3-ima.
12.3-ma.
n.a.naa.
2....na.
18.9.tn.a.
'LRRSR
ia.ina
62.3.ima..
9.9.ma.
na..na.
93.1
11.6dim..
TRDETVAD
Ina.
$7.9 1.4
9.1
La..
9.1
12.3:113
EGCN
75.6:10.8
64,3=0.9
28.12:1.0
74.3::111
99.1=0.2
96.92004
EGCN UR
75.7.008
27.51.022
73.9112
9.33
97.3110.4
EGCN- UR -TV
73.0-10.5
87.3-0.1
26.9-0.2
75.0-10.3
99.8=0.0
99.1-:0.1
GPN
68.2-26
8..=13
16.32:19
68.1=20.3
96.0=0.6
69.6210x
GPN- UR
62.8+333
92.2=0.6
36.9+2211
67.9-0.5
96.0=0.7
75.1#442
GPN -UR TT
6..6.1199
93.2
40.9
60.6.12.5
97.820.1
82.5+1111
Bold numbers indicate best model across all the uncertainty metrics each dataset. Underlined numbers the results the same model type for each dataset. n.a. means either model or metric not applicable. 'U/P-7' takes class 4 as the OOD class and 'UP. 8 class as the OOD
Table 15: OOD Detection Result for UH
Houston OOD ROC
Hosuton
dataset
OA
OOD PR
OA
OOD ROC
OOD PR
softmax-GCN
66.842.9
819920
46.4#09
67.7222
138433
5,0=0,2
GKDE
68.91n...
93.6-t..a.
68.74n..
68.64...a.
92.3....
66.5:tn..
RGAE
n.a.iaa.
85...n.a.
naa.naa.
46.0
7.72n..
TLRSR
n.a.na.
48.9....
6,3+na.
n.a.nn..
49.6.t..
8.0
EGCN
69.5.0.55
93.540.4
438811
70.242.33
96.6
70.1:4.3
EGCN UR
71.711.0
94.4110.1
47.51008
70.3.119
97.1
73.1
EGCN UR -TV
71.34-122
96.0-10.7
54.21162
70.54
97.01004
70.9:11.9
GPN
66.7.10.4
99.1110.3
8781722
64.5.144
96.5118
63.4113.44
GPN UR
66.14114
19:10:0.0
90.5+10.3
64.9-0.5
98.11:0.6
70.5:12.7
GPN -UR-TT -TV
66.3400.8
99.24:0.1
90.6+00.8
63.6
98.0++0.7
75.6#6.1
Bold numbers indicate the best model across all the uncertainty metrics for each datasct. Underlined numbers are the best results within the same model type for each dataset. n.a. means either model metric not applicable. "UH-O' takes class 0 as the OOD class and 'UP-1" Lakes class the OOD class.
Table 16: OOD Detection Result for UH (cont.)
Houston -10 OOD ROC
dataset
TD OA
0OD ROC
0OD PR
TD OA 73.4104
00D PR
sofimax GCN
65.9.177
62.0.199
4.510.5
73.2.114
14.7.10.6
GKDE
68.5ina.
32.3...aa.
56.9.nn..
71.4.a..
74.7.a..
32.0.in...
RGAE
n.a.an..
6384na
5.1=na..
na.ina.
24.6.a..
5.9m..
TLRSR
n.a.na.
49.4
3.4ina.
na.ina.
48.24n..
5.7ina
EGCN
70.0+008
90.6.0.11
16.740.
73.0::033
76.0:05
17.0:0.3
BGCN- UR
69.74116
91.0-2288
25.44777
73.14:0.1
78.3+:022
18.8+00.2
EGCN UR
70.4:0.3
97.311044
44.91442
73.11:0.6
77.110.44
18.1:10.3
GPN
64.1+0.3
70.14:06
6.1=0.1
70.9+0077
58.7+199
10.5
GPN UR
64.1
70.8:10.4
6.3-0.1
68.011.11
65.4-117
12.0
GPN -UR-U -TV
54.840.3
85.6++0.8
1188007
70.8
67.3+2.3
12.8#0.8
Bold numbers indicate the best model across all the uncertainty metrics for each datasct. Underlined numbers are the best results within same model type each
means either model or not applicable. 'UH-2' takes class OOD class "UP-10" takes class 10 as the ODD class.
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Table 17: OOD Detection Result for KSC
KSC-3
KSC-6
OA
OOD ROC
POD
OA
OOD ROC
OOD PR
solumax-GCI GCN
45.5EL1
GKDE
n..
47.0.in.a.
5.1
85.0..n.a.
43.2.tn..
2.2.m..a.
RGAE
naz
68.6d.a..
6.9
na.ina.
71.1aa.
4.1.ma.
TLRSR
67.7
6.0.ina.
na.ina.
7....n..
3.1.na..
TRDFTVAD
na
64.2 4.5
5.5 0.7
na-ina.
67. 17.14885
2.540.7
89.700.0
64.620.0
7.10000
87.6
11.4008
1920.0
EGCN- UR
89.6+:0.0
64.420011
7.0:0.0
87.6+0000
19.54:099
1,9=000
EGCN UR TT
90.0110.1
76.7.10.5
12.0.10.3
87.7..0.1
41.912888
2.210.11
66.8116
54.317.4
4.720.9
82.6E19
16.6
2,0=0.3
GPN UR
86.3:12222
$8.7122.6
0.0::033
80.71:0.9
86.71119
7.811:14
58.3.99.0
5.2.0.99
90.4
20.0.19.0
Bold numbers indicate the best model across all the uncertainly metrics for each dataset. Underlined numbers are the best results within same model type for each n.a. means either model metric applicable. 'KSC-5' 'KSC-6' takes class the OOD class.
Table 18: OOD Detection Result for KSC (cont.)
KSC-7
KSC-12
ID
OOD ROC
OOD PR
ID OA 80.9
OOD ROC
OOD PR
98.900.2
GKDE
85.8
88.94.n..
58.7in.a
100.0=2...
59.1..m...
RGAE
na.ina.
1861na.
4.8ma.
n.. ma..
93.01na.
56.6.1m..
TIRSR
n.a.ia..
67.9ina.
14.7ia..
na..na..
49.9.n..
9.I.n.a.
TRDFTVAD
nat.aa
70.74 2.7
22.64:
na..na..
9
9.8-0.11
EGCN
87.8:0.3
03.5:0033
51.13222
84.3=2011
100.0=0.00
99,9=0000
BOCN UR
87.710022
93.8-100
53.6:14.3
84.3:10.1
100.0::0.0
99.9:10.0
EGCN UR TV
87.910.11
93.7.10.8
56.2.16.2
84.2.10.1
100.0110.0
99.9.10.0
GPN
86.TEE
59.0:1..8
9.7:E18
100.010000
GPN UR
82.11..1
78.6.144
18.9112.7
77.410.9
100.010.0
99.9
81.61:66
79.54:1.6
20.64::.0
81.0-0.7
100.0:0.00
99.9
Bold numbers indicate the hest model across all the uncertainty metrics for each datnset. Underlined numbers are the best results within the same model type for each datasct. n.a. means cither model metric not applicable. 'KSC-7' takes class 7 as the class and KKSC 12' takes class as the OOD class.
Table 19: Ablation Study on EGCN2
UP- 6
UP-7
ID OA
ROC
PR
ID OA
ROC
PR
EGCN2
70.7=2.8
73,9+1.2
5.0=0.2
69.1
83.8=0.2
20.1
EGCN2 UR
65.3
84.6.10.3
7.410.1
63.7+2.7
85.5:10.1
22.0:10.2
EGCN2-TT TV
70.2
75.6:10.6
5.3+0.11
72.0+1144
84.0+10.1
20.4+10.2
EGCN2 - UR -TV
65.644.6
84.5+0.3
7.420.1
65.643.7
85,420.2
21.940.3
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F RELATED WORK
Hyperspectral Imaging Analysis. Due to the wealth of detailed spectral information available in each pixel, hyperspectral imaging (HSI) has found widespread application in various real-world scenarios. HSI classification (HSIC) aims to assign distinct class label to each pixel. In their work, (Chen et al. 2014) utilized stacked auto-encoder networks for HS image classification by leveraging dimensionally-reee
H$ images obtained through principal component analysis (PCA). Another study (Liu et al., 2017) introduced convolutional neural networks (CNNs) to effectively extract spatial-spectral features from HS images, resulting in improved classification performance. In sep- arate work, cascaded RNN was proposed (Hang al. 2019) to utilize spectral information com- prehensively for achieving high-aceuracy HS image classification. Furthermore, (Hong et al.) (020)) developed fusion modules that seamlessly integrate CNNS and miniGCNs in an end-to-end man- ner. We refer (Ahmad et al. 2021) for a complete review of HSI classification task. HSI spectral unmixing decompose the image into collection of reference spectral signatures with associated pro- portions, which is a non-negative matrix factorization problem (Lee & Seung 1999), which can be formed to blind and nonblind problem with linear mixing model (Qin al. (0200 or extended linear mixing model (Drumetz et al.) 2019). We refer to Bhatt & Joshi (2020) for systemic introduction. Anomaly detection on HSI involves detecting pixels an image whose spectral characteristics devi- ate significantly from the surrounding or overall background pixels and attracts a lot of interest Deep learning-based methods can be divided into CNN-based (Li et al., 2017). autoencoder-based (Bati et al. 2015), GAN based (Jiang et al.] 2020) and RNN based (Lyu & Lu 2016). Some other tech- niques include manifold learning (Lu et al. 2019) and low-rank representation (Xie et al. 2021
A &comprehensive review is conducted by Hu et al. 2022
Uncertainty quantification models on i..d inputs. Numerous studies have focused on develop- ing uncertainty quantification models for data that is independent of inputs, such as images. These efforts encompass various approaches, including multi-forward pass models such as ensembles [Lakshminarayanan et al.] (017), tropouu-based models (Gal & Ghahramani 2016), and deter- ministic models like Bayesian-based methods (Charpentier et al. 2022).
Uncertainty quantification on graph data. As pointed out in the survey (Abdar et al., 2021), uncertainty quantification on GNN and semi-supervised learning is under-explored. Most existing models for uncertainty quantification on graphs are either dropout-based or BNN-based methods that typically drop or assign probabilities to edges. Two approaches quantified uncertainty using deterministic single-past pass GNNs. One is called graph-based kernel Dirichlet distribution estimation (GKDE) (Zhao al. 2020), which consists of evidential GCN, graph-based kernel, teacher network, dropout, and loss regularization. Another method is the GPN (Stadle et al.) 2021) model that combines PN (Charpentier et al.) 2020) and personalized page rank (PPR) message passing to disentangle uncertainty with and without network effects. In addition, recent method (Wu et al.) 2023) used standard classification loss for OOD detection on graphs together with an energy function that is directly extracted from GNN. This method is limited to OOD detection, not generally on the topic of uncertainty quantification. However, these models have feature collapsing issues and the physical mixing properties of HSI are not yet built into the models.
Uncertainty quantification on hyperspectral image classification models. To the best of our knowledge, EGCN and GPN are the state-of-the-ard models on the node-level uncertainty estimation for graph data. Additionally, there is currently no directly related work on uncertainty quantification for hyperspectral imaging classification. As pointed out in a recent survey (Su et al. (02T). anomaly detection is one of the most closely related and well-studied topics in HSI. Therefore, we use three anomaly detection models as baselines. However, it's important to note that while anomaly detec- tion aims to detect pixels whose spectral characteristics deviate significantly from surrounding or background pixels, OOD detection focuses on identifying samples from distributions that are differ- ent from the (ID) training samples. Related work by He et al. (2022) investigated the confidence and calibration error for HSIC, which is subset of aleatoric uncertainty, but it can not detect OOD pix- els. Pal et al. (2022) examined the open-set recognition task, which does not provide any estimation of aleatoric uncertainty.
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Published as a conference paper at ICLR 2024
G LIMITATIONS
The UR term we introduce exhibits a strong correlation with the performance of unmixing. Specifi- cally, its effectiveness hinges on the accuracy the reference endmember matrix, which we assume is available in our problem formulation. Currently, we adopt the approach outlined in (Qin et al, 2020) for the endmember matrix, which serves as our reference. Nevertheless, a universally accepted criterion for assessing the quality of endmember matrices is lacking, and possible inaccurate one could potentially undermine the positive impact of our UR term. Besides, certain categories pose challenges for unmixing models due to unpredictable noise and intricate mixtures. In such instances, our proposed UR term might not offer substantial assistance. However, we introduce the uncertainty quantification problem in the HSI domain, which is less explored but necessary in real applications. Additionally, we propose promising direction in that we can improve deterministic uncertainty quantification models with domain knowledge, and the resulting insight obtained in this project can be extended to other application domains.
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Reviews:
[{"review_id": "9a1KidqE7t", "reviewer": "Reviewer_SyHe", "summary": "This paper analyzes the limitations of the ENN model based on UCE loss. Additionally, the author introduces two regularization terms to alleviate these constraints, and the proposed regularization terms exhibit significant improvements on some datasets.", "questions": "Please see the Weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "1. This article offers rich and comprehensive theoretical derivations, with strong formal proofs for the proposed regularization terms and UCE loss analysis, which makes the results quite convincing.\n2. The motivation behind the article is clear, and the analysis is praiseworthy.\n3. The proposed method has been tested on multiple benchmarks.\n4. The entire manuscript is well-structured and logically organized.", "weaknesses": "1. The experimental section of this manuscript appears to be less robust compared to the design and discussion of the methodology.\n2. The two proposed regularization terms, UR and TV, do not perform well in Table 1. I'm curious about the reasons behind this.\n3. There is a lack of comparative analysis in the experimental section. I'm interested in understanding how the results compare to the latest methods.\n4. The ablation experiments directly rely on the results from Tables 1 and 2 for analysis. Perhaps the author could benefit from incorporating more varied ablation designs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kZspmO9OJU", "reviewer": "Reviewer_pixa", "summary": "This paper introduces the uncertainty quantification to the Hyperspectral imaging classification (HSIC), i.e. epistemic and aleatoric uncertainties. Specifically, the paper theoretically analyzes the limitation of uncertainty cross-entropy (UCE) loss in evidential graph convolution neural networks (EGCN) and proposes two regularization terms to deal with this limitation. Experiments are performed on three real-world HSIC datasets.", "questions": "Could the author expand their analysis of the limitations and solutions related to the UCE model and apply it to the design of the EGCN model? This would provide a more coherent and insightful discussion.", "limitations": "", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. This paper analyzes the reason why the UCE-based MLP layer in EGCN cannot accurately obtain evidence predictions.\n2.  A solution for the above problem is also provided.", "weaknesses": "1. The rationality of the Unminxing-based Regularization part is insufficient. First, the HSIC problem containing noise will greatly affect the optimization results of Eq. (10). Secondly, even considering a simple case without noise, whether the ID class and OOD class can be split as Eq. (10) and its rationality remain to be verified.\n2. The unmixing-baed regularization (UR) term, i.e. Eq. (11), is designed heuristically, and its rationality has not been fully explained. Specifically, the optimization trends of $\\mathbf{b}^{i}(\\mathbf{\\theta})$ and $u^{i}(\\mathbf{\\theta})$ explained in the main text are consistent with the hyperspectral unmixing problem, but I think the rational explanation of this part is insufficient.\n3. The proposed evidence-based total variation regularization term is incremental. And Proposition 2 is a relatively simple and intuitive conclusion based on Eq. (13).\n4. The proposed improvements have not been consistently verified in experimental performance.\n5. The compared softax-GCN method is an earlier method, and the author can try some of the latest methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "R9W6CbSd2Q", "reviewer": "Reviewer_WkZQ", "summary": "This paper introduces a novel graph-based framework aimed at quantifying uncertainty in the field of HSIC. The authors analyze the limitations of the ENN models based on the UCE loss. To alleviate these limitations, this paper leverages the inherent physical properties of HS data and applies edge-preserving regularization techniques to facilitate the propagation of evidential information within the spatial domain, as a result, two regularization terms UR and evidence-based TV are proposed. The experiments on three datasets to demonstrate the effectiveness of the proposed regularizations.", "questions": "1.\tWhat does \"ID\" mean in this paper?\n2.\tIs 'Mis' an abbreviation for Misclassification in Table 1? It should be explained in the paper.\n3.\tDo the numbers following the dataset names in Table 11 to Table 16, such as '4' in 'UP-4,' represent the class numbers? It should be clarified in the paper for better understanding.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1.\tThis paper introduces a graph-based uncertainty quantification framework for HSIC and presents UR and TV regularization methods based on the characteristics of HS data.\n2.\tThe work on graph-based uncertainty quantification in the field of HSIC is novel and appears to be effective based on the provided experimental results.\n3.\tThe research is well-motivated, theoretically grounded, and explores a graph-based uncertainty quantification framework for HSIC models, aligning with the characteristics of hyperspectral imaging.", "weaknesses": "1.\tSome minor details in the writing require attention and revision. For instance, in the last paragraph of the Introduction, it should use the abbreviation 'TV' for 'total variation.' In the Conclusion, there is a shift from present tense to past tense, and it's essential to maintain consistency.\n2.\tIn Table 1, the proposed methods show Misclassification ROC scores lower than those of softmax-GCN on the UP and UH datasets, indicating that the best results have not been achieved. The paper lacks relevant explanations, and a more comprehensive and in-depth experimental analysis is needed.\n3.\tIn the comparative experiments, it's worth noting that there are instances where the introduction of TV leads to a decrease in results. For example, in Table 11 for OOD PR metric of UP-4, GPN-UR scores 99.1 and GPN-UR-TV scores 98.9. This raises questions about the robustness of these methods. It would be beneficial to provide relevant explanations and analyses. Additionally, it might be worthwhile to include separate experimental results for EGCN-TV and GPN-TV.\n4.\tThe compared baselines need to be enriched. There are some more recent methods published that should be introduced and compared, such as \" Hyperspectral Anomaly Detection Based on Tensor Ring Decomposition With Factors TV Regularization \", \" Hyperspectral anomaly detection based on variational background inference and generative adversarial network\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "66Aia6w34h", "reviewer": "Reviewer_NxUY", "summary": "The paper proposes two graph-based uncertainty quantification methods for hyperspectral image classification, incorporating advanced uncertainty quantification models (EGCN and GPN) and specific regularizations (unmixing and TV). The experimental results on three HS datasets demonstrate the advantages of the models. While the introduction of uncertainty quantification into HSIC is a valuable contribution, the overall novelty of the paper may be considered limited, and the experimental analysis could benefit from further depth and solidity.", "questions": "Please refer to the weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper serves as a pioneering work in the exploration of uncertainty estimation within graph-based HSIC models.\n2. The paper effectively presents a clear and well-defined motivation, along with a comprehensive theoretical analysis highlighting the limitations of prevalent uncertainty cross-entropy methods.\n3. The inclusion of informative unmixing-based and TV-based regularizations in the context of HSIC is noteworthy, and the results successfully confirm the effectiveness of these proposed designs.", "weaknesses": "1. Recently, there has been extensive exploration of Graph-based HSIC, demonstrating its effectiveness. However, a key challenge lies in the high computation and space complexity arising from the pixel-level graph. How do the authors address this issue?\n2. This paper's primary contribution lies in its incremental approach, integrating various existing works, including GNN, uncertainty, unmixing, and TV. Furthermore, the growing attention towards OOD and open-set recognition in HSI underscores the need for a compelling justification of the primary contribution.\n4. The experimental results, while promising, require additional substantiation to validate the efficacy of the proposed methods. A more comprehensive set of experiments is necessary. Furthermore, the authors should delve into a more detailed analysis of the experimental outcomes.\n5. Further elucidation is needed regarding the unmixing-based regularization. For instance, a detailed explanation of how the authors concurrently optimize abundance and endmembers within a graph net would enhance understanding.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "R1ktq8oGmo", "reviewer": "Reviewer_6WzU", "summary": "This manuscript adapted two advanced uncertainty quantification models EGCN and GPN for quantifying epistemic and aleatoric uncertainties associated with the HSIC results. Specifically, two regularization terms are proposed to mitigate the limitations of UCE loss function.", "questions": "What are the limitations of UCE and the specific role of UR? Please describe them directly and clearly in the Abstract and contribution section.\n\tFrom Table Ⅰ, it can be found that the proposed uncertainty quantification frameworks do not reach the SOTA level on the misclassification detection tasks, especially the GPN-based variants, what is the purpose of these experiments?\n\tHow computationally efficient is the use of UR and TV regularization terms on OOD and misclassification detection tasks.\n\tIt is suggested that authors unify the meaning of bold and underlined numbers in each table. In addition, some bold an underlined numbers in Table 11-15 are incorrect. Please check the corresponding issues through the full manuscript.\n\tIn the E.2-Detailed Result section, whether the proposed models are equally applicable to OOD detection for other classes in different datasets. Why are the PR values for class-10 in the UH dataset and class-7 in the KSC dataset below the baseline.\n\tThere are some punctuation errors that should not be made in a paper. For example, \"…64, 128, 256, respectively We…\" and \"…{10-4,10-3,10-2,10-1}(2) Number of… \"in part D-Model Details. It is suggested to carefully check the textual details of the manuscript. In addition, what do parameters λ and λ' represent in the model.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1.\tThis manuscript applied uncertainty quantification models to the new field of HSIC, eliminating the limitations of the uncertainty cross-entropy loss function.\n2.\tStudies on the uncertainty quantification of HSIC results are extremely rare, and the author's research direction is very interesting and meaningful.\n3.\tThis manuscript has provided clear questions and motivations, along with detailed formulae definitions and derivations.", "weaknesses": "1.\tThis manuscript is full of hyperparameters, and although the authors list the parameter choices for each model on each data set, the key parameters are not adequately discussed. For example, parametric sensitivity analysis of parameter β, λ1, λ2, and λ3 on each data set is necessary.\n2.\tThe ablation experiments in this manuscript are inadequate, for example, models using TV regularization term alone, not including UR regularization term.\n3.\tUncertainty has been widely studied in the previous works. This paper seems simply introduced the existed work for HSI classification. The experimental comparison is also insufficient.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)