H-SPLID: HSIC-based Latent Info Decomposition

• H-SPLID is a latent decomposition method that segregates predictive features (salient subspace) from irrelevant factors (non-salient subspace).
• It employs latent masking, geometric regularization, and HSIC penalties to enforce low-dimensional, task-relevant representations.
• Empirical results show improved robustness against adversarial and context-based perturbations across diverse datasets.

H-SPLID, short for HSIC-based Saliency Preserving Latent Information Decomposition, is a supervised representation-learning method that explicitly partitions a latent representation into a salient subspace and a non-salient subspace, with classification performed from the salient component only (Miklautz et al., 23 Oct 2025). Its central objective is to make a model rely on task-relevant information rather than background, context, or other nuisance structure. The method combines latent masking, geometric regularization, and HSIC-based dependence penalties to promote a low-dimensional predictive channel, and it is motivated by the observation that standard neural networks can achieve high clean accuracy while still encoding a mixture of relevant and irrelevant factors in one entangled latent space (Miklautz et al., 23 Oct 2025).

1. Problem setting and conceptual basis

H-SPLID targets classification problems in which input signals contain both label-relevant and label-irrelevant variation. The motivating claim is that ordinary training often yields representations that are predictive but not genuinely salient: a classifier may succeed on clean data while still depending on backgrounds, co-occurring context, or redundant visual structure (Miklautz et al., 23 Oct 2025). In the paper’s diagnostic example, an image contains two MNIST digits side by side while the label is the left digit only; a vanilla classifier retains high clean accuracy yet suffers a large accuracy drop when the right digit is adversarially perturbed, which is taken as evidence that the learned representation is not restricted to the intended signal (Miklautz et al., 23 Oct 2025).

Within this framework, “saliency preserving” does not denote a post hoc attribution map. It denotes a representational constraint: the latent space is organized so that the information salient for predicting $Y$ is preserved in one subspace, while remaining variability is pushed into another (Miklautz et al., 23 Oct 2025). The method is therefore positioned simultaneously as a robustness method, an interpretability method, and a task-relevant representation-learning method.

The paper states that H-SPLID does not require external saliency annotations or a separate background dataset (Miklautz et al., 23 Oct 2025). This places it in a regime distinct from methods that depend on pixel-level masks, environment labels, or explicitly curated nuisance datasets.

2. Latent decomposition and objective function

The model considers $k$-class classification on a dataset

$$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$

with $\mathcal{X}\subseteq \mathbb{R}^d$ and $\mathcal{Y}\subseteq \mathbb{R}^k$, where each $y_i$ is one-hot encoded (Miklautz et al., 23 Oct 2025). The network consists of an encoder

$f_\psi:\mathcal{X}\to \mathbb{R}^m$

and a linear output layer

$g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$

so that the full predictor is $h_\theta=g_W\circ f_\psi$ with $\theta=\{\psi,W\}$ (Miklautz et al., 23 Oct 2025).

The central mechanism is a diagonal mask

$k$0

with complement

$k$1

Given $k$2, H-SPLID defines

$k$3

Only $k$4 is fed to the classifier (Miklautz et al., 23 Oct 2025). The non-salient component is not discarded; it is explicitly used in the training objective so that it absorbs variation that should not drive prediction.

Two centroid-based geometric terms organize the latent space. With class index sets $k$5, class centroids $k$6, and global centroid $k$7, the paper uses

$k$8

and

$k$9

The intended interpretation is that $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$0 promotes within-class clustering in the salient space, while $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$1 promotes global alignment in the non-salient space (Miklautz et al., 23 Oct 2025).

The information-theoretic component is expressed through two HSIC penalties: one between the input and the salient representation, $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$2, and one between the labels and the non-salient representation, $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$3 (Miklautz et al., 23 Oct 2025). The first is used to reduce redundant dependence of $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$4 on the full input; the second is used to discourage label information from leaking into $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$5.

The full objective is

$$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$6

where $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$7 is a masked cross-entropy loss using only the salient coordinates (Miklautz et al., 23 Oct 2025). The paper characterizes the combined effect as enforcing prediction from $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$8 only, reducing redundancy within $$\mathcal{D}=\{(x_i,y_i)\}_{i=1}^n \subseteq \mathcal{X}\times\mathcal{Y},$$9, and removing class-relevant signal from $\mathcal{X}\subseteq \mathbb{R}^d$0.

3. Optimization procedure and mask mechanics

H-SPLID is trained by alternating optimization over network parameters and the salient mask (Miklautz et al., 23 Oct 2025). With the mask fixed, the network parameters $\mathcal{X}\subseteq \mathbb{R}^d$1 are updated for one epoch using minibatch SGD on the full objective. For each minibatch, the algorithm computes latent codes, minibatch class means, the global mean, the H-SPLID loss, and then applies a gradient step (Miklautz et al., 23 Oct 2025).

With $\mathcal{X}\subseteq \mathbb{R}^d$2 fixed, the mask is updated by minimizing the geometric part of the objective,

$\mathcal{X}\subseteq \mathbb{R}^d$3

subject to $\mathcal{X}\subseteq \mathbb{R}^d$4 and $\mathcal{X}\subseteq \mathbb{R}^d$5 (Miklautz et al., 23 Oct 2025). The paper states that this yields a closed-form coordinate update for $\mathcal{X}\subseteq \mathbb{R}^d$6, interpreted as follows: a latent dimension that varies strongly at the global level but weakly within classes is assigned higher saliency weight, whereas a dimension that mainly reflects global or shared variation is pushed toward the non-salient side (Miklautz et al., 23 Oct 2025). The optimization produces a continuous mask $\mathcal{X}\subseteq \mathbb{R}^d$7, which is thresholded at $\mathcal{X}\subseteq \mathbb{R}^d$8 at the end, although the reported behavior is that many coordinates already lie near $\mathcal{X}\subseteq \mathbb{R}^d$9 or $\mathcal{Y}\subseteq \mathbb{R}^k$0 (Miklautz et al., 23 Oct 2025).

To stabilize training, the paper uses a moving-average update for the mask and initializes the procedure with $\mathcal{Y}\subseteq \mathbb{R}^k$1 (Miklautz et al., 23 Oct 2025). For ImageNet-scale experiments, the implementation introduces $\mathcal{Y}\subseteq \mathbb{R}^k$2 and $\mathcal{Y}\subseteq \mathbb{R}^k$3, which specify the fraction of the data used for mask initialization and the fraction processed before each mask update (Miklautz et al., 23 Oct 2025).

This alternating scheme induces moderate overhead. On COCO, the reported average epoch times are $\mathcal{Y}\subseteq \mathbb{R}^k$4 s for Vanilla, $\mathcal{Y}\subseteq \mathbb{R}^k$5 s for HBaR, and $\mathcal{Y}\subseteq \mathbb{R}^k$6 s for H-SPLID (Miklautz et al., 23 Oct 2025). On ImageNet-1K, total runtimes for 20 epochs are reported as 11h27m for Vanilla, 12h04m for HBaR, and 14h06m for H-SPLID (Miklautz et al., 23 Oct 2025). The paper attributes the overhead to HSIC computation and mask updates.

4. Theoretical properties

The main theoretical claim is a robustness bound linking perturbation sensitivity to both the dimension of the salient subspace and the HSIC between inputs and salient representations (Miklautz et al., 23 Oct 2025). Let

$\mathcal{Y}\subseteq \mathbb{R}^k$7

where $\mathcal{Y}\subseteq \mathbb{R}^k$8 is the number of active salient dimensions. Under the assumptions that $\mathcal{Y}\subseteq \mathbb{R}^k$9 is sampled from a truncated multivariate normal, the encoder $y_i$0 is $y_i$1-Lipschitz, the output layer is linear and bounded, and the kernels used for HSIC are continuous, positive definite, universal on compact domains, and uniformly bounded in sup norm, the paper proves that the expected prediction deviation under bounded perturbations is upper-bounded by a term that scales with $y_i$2 and $y_i$3 (Miklautz et al., 23 Oct 2025).

The conceptual interpretation is explicit: reducing the dimensionality of the predictive channel and reducing the amount of input information preserved in that channel both improve robustness (Miklautz et al., 23 Oct 2025). The paper presents this as a robustness–compression link rather than as a purely empirical regularization effect.

A corollary bounds the probability of encountering an input at which the Frobenius norm of the input gradient exceeds a threshold. In the paper’s formulation, the measure of the “salient-active” region is controlled by the same structural quantities: salient-space dimension and $y_i$4 (Miklautz et al., 23 Oct 2025). The proof sketch proceeds by bounding network outputs using salient sparsity, relating bounded neural functions to RKHS function classes, using HSIC to control covariances, applying a Stein-type identity for truncated Gaussian inputs, and then converting the gradient bound into a perturbation bound through a first-order Taylor argument (Miklautz et al., 23 Oct 2025).

The theory is not presented as a general theorem for arbitrary data distributions. The assumptions are specific, and the paper treats them as the analytic setting under which the robustness–compression relationship can be formalized (Miklautz et al., 23 Oct 2025).

5. Empirical results

The empirical evaluation covers five datasets: C-MNIST, a four-class COCO subset, ISIC-2017, ImageNet-9, and CounterAnimal (Miklautz et al., 23 Oct 2025). Baselines include Vanilla cross-entropy, weight decay, $y_i$5 sparse weights, $y_i$6 sparse activations, group-lasso weights, group-lasso activations, and HBaR (Miklautz et al., 23 Oct 2025). All methods are trained on clean data only, with no adversarial training and no saliency masks used during training (Miklautz et al., 23 Oct 2025).

On the diagnostic C-MNIST task, the paper reports that Vanilla achieves clean accuracy $y_i$7 but only $y_i$8 under strong PGD applied to the irrelevant right digit, whereas H-SPLID achieves clean accuracy $y_i$9 and $f_\psi:\mathcal{X}\to \mathbb{R}^m$0 under the same attack (Miklautz et al., 23 Oct 2025). The non-salient space alone attains only $f_\psi:\mathcal{X}\to \mathbb{R}^m$1 classification accuracy, which the paper interprets as evidence that class information has largely been removed from that channel (Miklautz et al., 23 Oct 2025).

On COCO, the most direct evaluation concerns attacks on backgrounds. Under background-only PGD at $f_\psi:\mathcal{X}\to \mathbb{R}^m$2, the reported accuracies are $f_\psi:\mathcal{X}\to \mathbb{R}^m$3 for Vanilla, $f_\psi:\mathcal{X}\to \mathbb{R}^m$4 for weight decay, $f_\psi:\mathcal{X}\to \mathbb{R}^m$5 for HBaR, and $f_\psi:\mathcal{X}\to \mathbb{R}^m$6 for H-SPLID (Miklautz et al., 23 Oct 2025). Under full-image PGD at $f_\psi:\mathcal{X}\to \mathbb{R}^m$7, Vanilla attains $f_\psi:\mathcal{X}\to \mathbb{R}^m$8, HBaR $f_\psi:\mathcal{X}\to \mathbb{R}^m$9, and H-SPLID $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$0 (Miklautz et al., 23 Oct 2025). Under background-only AutoAttack at $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$1, Vanilla is $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$2, HBaR $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$3, and H-SPLID $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$4; under full-image AutoAttack at $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$5, Vanilla is $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$6, HBaR $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$7, and H-SPLID $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$8 (Miklautz et al., 23 Oct 2025). These results are presented as evidence that H-SPLID reduces dependence on non-salient image regions.

On ISIC-2017, where perturbations affect non-lesion pixels, H-SPLID again yields the best reported performance: $g_W(z)=Wz,\qquad W\in\mathbb{R}^{k\times m},$9 without perturbation, $h_\theta=g_W\circ f_\psi$0 under brightness perturbation, $h_\theta=g_W\circ f_\psi$1 under defocus blur, and $h_\theta=g_W\circ f_\psi$2 under occlusion-like corruption; the corresponding HBaR values are $h_\theta=g_W\circ f_\psi$3, $h_\theta=g_W\circ f_\psi$4, $h_\theta=g_W\circ f_\psi$5, and $h_\theta=g_W\circ f_\psi$6 (Miklautz et al., 23 Oct 2025). The paper treats this as evidence that the decomposition is not limited to synthetic backgrounds.

On ImageNet-9 and CounterAnimal, the reported values are $h_\theta=g_W\circ f_\psi$7 on IN-9 original, $h_\theta=g_W\circ f_\psi$8 on Only-FG, $h_\theta=g_W\circ f_\psi$9 on MixedRand, $\theta=\{\psi,W\}$0 on CA-Common, and $\theta=\{\psi,W\}$1 on CA-Counter for H-SPLID (Miklautz et al., 23 Oct 2025). The gains on MixedRand and CA-Counter are emphasized because these settings expose reliance on spurious background context (Miklautz et al., 23 Oct 2025).

The ablation study on COCO separates the contributions of geometry and HSIC. The paper reports $\theta=\{\psi,W\}$2 background-attack and $\theta=\{\psi,W\}$3 full-attack accuracy for cross-entropy only, $\theta=\{\psi,W\}$4 and $\theta=\{\psi,W\}$5 for cross-entropy plus $\theta=\{\psi,W\}$6, $\theta=\{\psi,W\}$7 and $\theta=\{\psi,W\}$8 for cross-entropy plus HSIC only, and $\theta=\{\psi,W\}$9 and $k$00 for the full H-SPLID objective (Miklautz et al., 23 Oct 2025). It therefore argues that neither ingredient alone accounts for the observed behavior.

The paper also reports evidence for low-dimensional saliency. On COCO, one of the best configurations yields a salient subspace of 14 out of 512 dimensions (Miklautz et al., 23 Oct 2025). Increasing $k$01 or $k$02 tends to shrink the salient space and improve robustness up to a point, while excessive regularization can collapse the model to trivial solutions (Miklautz et al., 23 Oct 2025).

6. Interpretation, limitations, and disambiguation

H-SPLID is best understood as a task-directed latent partitioning method rather than as a generic sparsification scheme. Compared with disentanglement methods, it is not trying to recover independent generative factors in an unsupervised sense; “salient” is defined operationally as useful for the label (Miklautz et al., 23 Oct 2025). Compared with invariant learning, it seeks insensitivity to nuisance factors through explicit latent partitioning and HSIC regularization rather than through environment annotations or causal invariance assumptions (Miklautz et al., 23 Oct 2025). Compared with information bottleneck-style approaches such as HBaR, it imposes distinct pressures on the two latent subspaces: it compresses input dependence in $k$03, removes label information from $k$04, and assigns geometric roles to both spaces (Miklautz et al., 23 Oct 2025). Compared with $k$05 or group-lasso regularization, it is not merely making weights or activations small; it is attempting to determine which latent dimensions should carry discriminative information and which should not (Miklautz et al., 23 Oct 2025).

The method’s stated applicability is strongest when irrelevant information is present in the input and the dataset is diverse enough that salient and non-salient factors do not always co-occur (Miklautz et al., 23 Oct 2025). The paper explicitly notes that if every class always appears in the same context, the model cannot disentangle object from background from labels alone; in that case additional supervision or prior knowledge would be needed (Miklautz et al., 23 Oct 2025). Other caveats identified in the paper are the computational overhead, the image-centric experimental scope, the distributional and kernel assumptions required by the theory, and the possibility of mask collapse under poor hyperparameter settings (Miklautz et al., 23 Oct 2025).

The acronym also benefits from disambiguation. In current arXiv usage, H-SPLID most directly refers to HSIC-based Saliency Preserving Latent Information Decomposition (Miklautz et al., 23 Oct 2025). Other supplied papers discuss unrelated methods that have been suggested as possible intended matches when the string is used ambiguously, including the Hierarchical Hamiltonian Splitting family for collisionless $k$06-body simulation (Aguilar-Argüello et al., 2020) and the HerMES prior-based SPIRE deblending and cross-identification framework associated with the HerMES XID algorithm (Roseboom et al., 2010). Those methods are distinct in domain, formalism, and purpose.