KPHD-Net: Uncertainty-Aware Multi-View Learning

• The paper introduces KPHD-Net, a framework that replaces conventional KL divergence with Proper Hölder Divergence for improved uncertainty quantification in incomplete multi-view data.
• It employs evidential Dirichlet modeling and Dempster–Shafer theory fusion to reduce the impact of corrupted or missing views on classification and clustering outcomes.
• Integration of Kalman filtering for temporal smoothing and variational regularization demonstrates enhanced stability and performance across diverse multi-modal datasets.

to=arxiv_search code ลุ้นบาท {"query":"(Xue et al., 14 Jul 2025) KPHD-Net Uncertainty Quantification for Incomplete Multi-View Data Using Divergence Measures", "max_results": 5} to=arxiv_search code ացինյալ {"query":"(Xue et al., 14 Jul 2025)", "max_results": 10} KPHD-Net is a multi-view learning framework for classification and clustering under incomplete, corrupted, or noisy views, introduced in "Uncertainty Quantification for Incomplete Multi-View Data Using Divergence Measures" (Xue et al., 14 Jul 2025). It targets two coupled failure modes in realistic multi-modal settings: unreliable uncertainty quantification when some views are missing or degraded, and brittle cross-view alignment when modalities exhibit domain gaps. The method replaces Kullback–Leibler divergence with Proper Hölder Divergence (PHD), models per-view class probabilities with variational Dirichlet distributions derived from evidential outputs, fuses view-specific beliefs with Dempster–Shafer theory (DST), and applies a Kalman filter to smooth fused beliefs and provide future state estimates. In the reported formulation, reliability is assessed through evidential uncertainty at the view level and through conflict and uncertainty masses during DST fusion, so corrupted or missing views contribute proportionally less to the final decision (Xue et al., 14 Jul 2025).

1. Problem setting and design rationale

KPHD-Net is situated in incomplete multi-view learning, where multiple modalities or views are available in principle, but one or more views may be absent, corrupted, or noisy at training or inference time. The motivating examples include RGB-D settings, and the paper emphasizes that two difficulties compound each other in practice: quantifying uncertainty reliably under incomplete or noisy views, and measuring cross-view distribution discrepancies under domain gaps.

The stated motivation for replacing KL divergence is specific. Conventional KL is described as asymmetric, sensitive to support mismatch, and prone to biased regularization when modalities follow multi-modal or heavy-tailed distributions. KPHD-Net addresses this by substituting Proper Hölder Divergence for distribution alignment and regularization, using evidential Dirichlet modeling for uncertainty-aware class-probability estimation, DST for conflict-aware fusion, and Kalman smoothing for stabilizing the fused output over time or sample sequences (Xue et al., 14 Jul 2025).

The framework is intended for both supervised multi-view classification and unsupervised multi-view clustering. This dual scope is notable because the same evidential and divergence-based machinery is used across discriminative and unsupervised settings. A plausible implication is that the authors view uncertainty-aware fusion as a general-purpose substrate for multi-view inference, rather than a task-specific add-on.

2. Architecture and incomplete-view handling

KPHD-Net is presented as an end-to-end framework with four principal components: per-view encoders, an evidential modeling block, a fusion mechanism based on DST and Kalman filtering, and a PHD-based divergence learning module (Xue et al., 14 Jul 2025).

Each view $v \in \{1,\dots,V\}$ is processed by a backbone network. The details explicitly mention ResNet50 as an example and note that UNet, DenseNet, ViT, and Mamba are also considered in experiments. Instead of a standard softmax head, each encoder produces per-class non-negative activations interpreted as evidence. This architectural choice is essential because evidential learning requires non-negative evidence $e^v \in \mathbb{R}_+^K$, which is then converted into Dirichlet concentration parameters by

$\alpha^v = e^v + 1$

component-wise.

The evidential block maps these concentrations into belief masses and uncertainty via subjective logic. Fusion is then performed at the belief level rather than at the raw feature or probability level. The fused belief masses, or fused “true” belief signal, are subsequently passed to a Kalman filter for temporal smoothing and future state estimation.

Handling incomplete views is an explicit design target. Missing views are simulated through channel removal or missing-rate masks, and corrupted inputs through Gaussian noise. The mechanism for robustness is not based on hand-crafted imputation. Instead, a missing or corrupted view is expected to produce low evidence and therefore high uncertainty mass $u^v$, which reduces its contribution during DST fusion. Conflict among views is handled directly by the DST combination rule, so contradictory beliefs do not dominate the fused decision. The framework can also introduce pseudo-views, constructed by concatenating outputs of two backbones, to stabilize learning when a view is missing. This suggests a design preference for uncertainty-weighted evidence aggregation over explicit reconstruction of absent modalities.

3. Proper Hölder divergence and variational regularization

A central technical feature of KPHD-Net is the use of Proper Hölder Divergence in place of KL divergence. For conjugate exponents $\alpha,\beta>0$ satisfying

$\frac{1}{\alpha} + \frac{1}{\beta} = 1$

and parameter $\gamma>0$, the Proper Hölder Divergence between densities $p(x)$ and $q(x)$ is given as

$$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$

The paper attributes three advantages to PHD relative to KL: it is symmetricizable and tunable through $e^v \in \mathbb{R}_+^K$0, it admits closed-form expressions for conic and affine exponential families including the Dirichlet family, and it measures discrepancies more effectively when modalities have heterogeneous statistics (Xue et al., 14 Jul 2025). For distributions in the same exponential family with natural parameters $e^v \in \mathbb{R}_+^K$1 and $e^v \in \mathbb{R}_+^K$2 and log-normalizer $e^v \in \mathbb{R}_+^K$3, the divergence is given in closed form by

$e^v \in \mathbb{R}_+^K$4

A symmetric form is also stated: $e^v \in \mathbb{R}_+^K$5

For Dirichlet distributions, the paper treats the family as an exponential family with natural parameter $e^v \in \mathbb{R}_+^K$6 component-wise and log-normalizer

$e^v \in \mathbb{R}_+^K$7

Gradients are obtained from

$e^v \in \mathbb{R}_+^K$8

where $e^v \in \mathbb{R}_+^K$9 is the digamma function.

In the variational objective, KPHD-Net replaces the KL term in a VAE-style ELBO by PHD: $\alpha^v = e^v + 1$0 The paper argues that this replacement yields tighter and more flexible regularization, mitigates mode-collapse, and better captures multi-modal or domain-gap discrepancies. Because the claims are presented as theoretical statements in the paper, the strongest warranted summary is that the method is analytically motivated by the closed-form structure of PHD and the strict convexity of $\alpha^v = e^v + 1$1, rather than by a purely empirical substitution.

4. Evidential Dirichlet modeling and DST-Kalman fusion

Per-view class probabilities are represented by variational Dirichlet distributions. For concentration vector $\alpha^v = e^v + 1$2, the Dirichlet density is

$\alpha^v = e^v + 1$3

For each view $\alpha^v = e^v + 1$4 and class $\alpha^v = e^v + 1$5, KPHD-Net defines non-negative evidence $\alpha^v = e^v + 1$6, concentration $\alpha^v = e^v + 1$7, total evidence

$\alpha^v = e^v + 1$8

and Dirichlet mean

$\alpha^v = e^v + 1$9

Uncertainty is measured by

$u^v$0

and belief mass for class $u^v$1 by

$u^v$2

These satisfy

$u^v$3

When probabilities are needed, a uniform base rate $u^v$4 is used, yielding

$u^v$5

DST provides the fusion layer. For a frame of discernment $u^v$6, a basic probability assignment $u^v$7 maps subsets $u^v$8 to $u^v$9 with $\alpha,\beta>0$0. Dempster’s combination rule for two BPAs $\alpha,\beta>0$1 is

$\alpha,\beta>0$2

where $\alpha,\beta>0$3 is the conflict. In the simplified singletons-only fusion used in the paper, the classwise fused belief and fused uncertainty are

$\alpha,\beta>0$4

with conflict $\alpha,\beta>0$5 computed from mutually exclusive singleton beliefs, approximately

$\alpha,\beta>0$6

in the singletons-only case. For more than two views, fusion is applied iteratively: $\alpha,\beta>0$7

The Kalman filter is then used to smooth the fused DST outputs. With state $\alpha,\beta>0$8, control $\alpha,\beta>0$9, measurement $\frac{1}{\alpha} + \frac{1}{\beta} = 1$0, system matrices $\frac{1}{\alpha} + \frac{1}{\beta} = 1$1, and covariances $\frac{1}{\alpha} + \frac{1}{\beta} = 1$2, the paper gives the standard linear prediction and update equations: $\frac{1}{\alpha} + \frac{1}{\beta} = 1$3

$\frac{1}{\alpha} + \frac{1}{\beta} = 1$4

$\frac{1}{\alpha} + \frac{1}{\beta} = 1$5

The role of the filter is to treat the fused DST “true” mass or fused classwise masses as measurements $\frac{1}{\alpha} + \frac{1}{\beta} = 1$6, thereby reducing jitter due to fluctuating view quality. The details specify that higher uncertainty $\frac{1}{\alpha} + \frac{1}{\beta} = 1$7 increases the measurement noise $\frac{1}{\alpha} + \frac{1}{\beta} = 1$8, while $\frac{1}{\alpha} + \frac{1}{\beta} = 1$9 controls expected temporal variation of beliefs. This indicates that uncertainty is not only used for static fusion weighting, but also propagated into the temporal smoothing stage.

5. Objectives, optimization, and computational profile

KPHD-Net optimizes a composite objective combining fused-view classification, per-view classification, and pseudo-view regularization. For sample $\gamma>0$0 with views $\gamma>0$1 and label $\gamma>0$2, the overall loss is

$\gamma>0$3

The fused-view term is

$\gamma>0$4

the pseudo-view term is

$\gamma>0$5

and each per-view term is

$\gamma>0$6

The expectation may be approximated by the Dirichlet mean $\gamma>0$7, and the class likelihood is stated as $\gamma>0$8. The paper also writes cross-entropy explicitly as

$\gamma>0$9

with $p(x)$0 optionally taken as fused probabilities $p(x)$1.

Optimization uses Adam with weight decay and learning-rate decay in PyTorch; $p(x)$2 for the divergence are tuned by grid search, and $p(x)$3 is a regularization weight. The text states that a curriculum or annealing strategy is not mandated, and that $p(x)$4 can be fixed or scheduled. The high-level algorithm is straightforward: encode each view into evidence, form Dirichlet parameters and subjective-logic masses, fuse via DST, optionally construct pseudo-views, compute per-view, fused, and pseudo losses with PHD regularization against a uniform Dirichlet prior, and update the parameters with Adam.

The reported computational scaling is also explicit. Encoder cost is dominated by the underlying backbones, written as $p(x)$5 per sample. Dirichlet and PHD computation requires $p(x)$6 per sample for per-view terms plus $p(x)$7 for fused terms, because evaluating $p(x)$8 involves $p(x)$9 Gamma or log-Gamma operations. Singletons-only DST fusion is $q(x)$0, while a full $q(x)$1-dimensional Kalman filter incurs $q(x)$2 in the worst case due to matrix inversion, though scalar or small-state smoothing reduces this cost.

6. Experimental results, ablations, and limitations

The reported experiments cover both classification and clustering (Xue et al., 14 Jul 2025). Classification datasets are SUNRGBD with 19 classes, NYUDv2 with 10 reorganized classes, ADE20K with 10 classes as an RGB-D subset, and ScanNet with 21 categories. Clustering datasets are MSRC-V1 with 7 classes, ORL with 40 subjects, MNIST using a 4K subset, Caltech101-7, and Caltech101-20. Missing and corrupted views are simulated using Gaussian noise with $q(x)$3 and missing rates $q(x)$4 by channel removal. The implementation uses PyTorch, Adam, dual 4090 GPUs, inputs resized to $q(x)$5 and cropped to $q(x)$6, and pseudo-views formed from concatenated backbone outputs.

Selected classification results emphasize fusion accuracy and robustness. On ADE20K with missing views $q(x)$7 and $q(x)$8, the fusion ACC values are reported as $q(x)$9, $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$0, and $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$1, with fusion F1 scores $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$2, $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$3, and $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$4; the ETMC fusion ACC is listed as $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$5. On NYUDv2, the best fusion setting $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$6 yields fusion ACC $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$7 versus ETMC $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$8, with per-modality RGB and Depth both at $$D^{\mathrm{H}}_{\alpha,\gamma}(p:q) = -\log\left( \frac{ \int_{\mathcal{X}} p(x)^{\gamma/\alpha}\, q(x)^{\gamma/\beta}\, \mathrm{d}x }{ \left(\int_{\mathcal{X}} p(x)^\gamma\, \mathrm{d}x\right)^{1/\alpha} \left(\int_{\mathcal{X}} q(x)^\gamma\, \mathrm{d}x\right)^{1/\beta} } \right).$$9. On ScanNet with $e^v \in \mathbb{R}_+^K$00, fusion ACC is $e^v \in \mathbb{R}_+^K$01 versus ETMC $e^v \in \mathbb{R}_+^K$02, and RGB ACC is reported up to $e^v \in \mathbb{R}_+^K$03. On SUNRGB-D with $e^v \in \mathbb{R}_+^K$04, fusion ACC is $e^v \in \mathbb{R}_+^K$05 versus ETMC $e^v \in \mathbb{R}_+^K$06, and depth ACC is $e^v \in \mathbb{R}_+^K$07.

Noise robustness is highlighted on ADE20K. With ETMC baseline fusion ACC $e^v \in \mathbb{R}_+^K$08, KPHD-Net reaches fusion ACC up to $e^v \in \mathbb{R}_+^K$09 at $e^v \in \mathbb{R}_+^K$10 using $e^v \in \mathbb{R}_+^K$11, close to the no-noise $e^v \in \mathbb{R}_+^K$12, and up to $e^v \in \mathbb{R}_+^K$13 at $e^v \in \mathbb{R}_+^K$14 using $e^v \in \mathbb{R}_+^K$15, again consistently surpassing ETMC.

The clustering results are particularly strong in the reported setting. Fusion clustering accuracy is $e^v \in \mathbb{R}_+^K$16 on MSRC-V1, compared with DSRL $e^v \in \mathbb{R}_+^K$17; $e^v \in \mathbb{R}_+^K$18 on ORL, compared with DSRL $e^v \in \mathbb{R}_+^K$19; $e^v \in \mathbb{R}_+^K$20 on MNIST, outperforming MCGC $e^v \in \mathbb{R}_+^K$21 and MCBVA $e^v \in \mathbb{R}_+^K$22; $e^v \in \mathbb{R}_+^K$23 on Caltech101-7, outperforming DSRL $e^v \in \mathbb{R}_+^K$24 and DSCMC $e^v \in \mathbb{R}_+^K$25; and $e^v \in \mathbb{R}_+^K$26 on Caltech101-20, beating DSRL $e^v \in \mathbb{R}_+^K$27 and described as close to DIMvLN $e^v \in \mathbb{R}_+^K$28 in a different setting.

The ablation studies separate the contribution of the divergence and evidential design. Across ADE20K, NYUDv2, and SUNRGB-D, replacing KL with Hölder yields consistent gains; the example given is ADE20K fusion at $e^v \in \mathbb{R}_+^K$29 versus $e^v \in \mathbb{R}_+^K$30. Hölder plus Dirichlet often further improves fusion on structured datasets, with ADE20K up to $e^v \in \mathbb{R}_+^K$31, although the paper notes small trade-offs on noisier datasets. Sensitivity analysis on ADE20K identifies optimal ranges $e^v \in \mathbb{R}_+^K$32 and $e^v \in \mathbb{R}_+^K$33, while excessively large $e^v \in \mathbb{R}_+^K$34 degrade performance through over-regularization. In a backbone comparison for Caltech101-20 clustering, DenseNet achieves best fusion $e^v \in \mathbb{R}_+^K$35, ResNet50 attains $e^v \in \mathbb{R}_+^K$36, UNet and Mamba underperform, and ViT gives strong RGB performance but slightly lower fusion than DenseNet.

The paper’s discussion of limitations is comparatively concrete. Performance is sensitive to the PHD hyperparameters $e^v \in \mathbb{R}_+^K$37, so grid search is recommended. Computing Gamma and log-Gamma terms for Dirichlet distributions and performing DST fusion introduce modest overhead relative to plain softmax classifiers. The DST implementation uses a singletons-only simplification for efficiency and therefore does not exploit composite focal elements; the text states that richer DST rules may improve some cases. The implementation guidance reflects these constraints: use non-negative activations for evidence, monitor $e^v \in \mathbb{R}_+^K$38 and $e^v \in \mathbb{R}_+^K$39 for calibration, initialize PHD around $e^v \in \mathbb{R}_+^K$40 and $e^v \in \mathbb{R}_+^K$41, set $e^v \in \mathbb{R}_+^K$42 depending on dataset noise, and choose Kalman measurement noise $e^v \in \mathbb{R}_+^K$43 proportional to fused uncertainty $e^v \in \mathbb{R}_+^K$44.

Taken together, the reported results support a specific interpretation of KPHD-Net: it is not only a divergence substitution, but an integrated evidential multi-view pipeline in which divergence regularization, subjective-logic uncertainty, DST conflict handling, and Kalman smoothing are all operative. A common misunderstanding would be to view the method as a conventional multi-view classifier with post hoc uncertainty estimation. The formulation and experiments instead place uncertainty at the center of representation, fusion, and temporal stabilization throughout the model (Xue et al., 14 Jul 2025).