Multi-Granular Discretization (IG-MD)

• Multi-Granular Discretization (IG-MD) is a dual framework that treats discretization via a geometric OT–IG approach for transport cost analysis and a multi-resolution method for intrusion detection.
• In the OT–IG formulation, discretization is encoded using second-moment matrices, Jacobian projections, and Frobenius decompositions to precisely quantify transport-information cost.
• The intrusion detection approach uses multi-granular Gaussian tokenization to generate coherent symbolic patterns, enhancing interpretability and improving detection accuracy.

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to=search_arxiv 公众号天天中彩票 ացնում 的天天彩票 天天中彩票中 Multi-Granular Discretization (IG-MD) is an arXiv label currently used for two distinct discretization-centered frameworks. In one usage, IG-MD denotes a geometric treatment of discretization-induced transport-information cost, where discretization of a continuous support is encoded by the second-moment matrix of a cell $\omega$ and analyzed through the correspondence between optimal transport (OT) and information geometry (IG), including the rank-deficient case (Yuge, 19 May 2026). In the other, IG-MD denotes a multi-resolution symbolic encoding for Interpretable Generalization in intrusion detection, where each continuous feature is represented at several Gaussian-based resolutions and supplied to a coherent-pattern rule learner (Chung et al., 16 Jul 2025). The shared acronym therefore names two technical programs with different state spaces, objectives, and evaluation criteria.

1. Scope and terminological usage

Recent arXiv usage assigns the name IG-MD to two formulations that both revolve around discretization but do so in different mathematical settings. The first treats discretization as a geometric operation on continuous probability distributions and studies the induced transport-information cost under identifiable and rank-deficient parametrizations. The second treats discretization as a preprocessing and representation mechanism for explainable intrusion detection, where multiple symbolic resolutions are used to improve rule quality and preserve auditability (Yuge, 19 May 2026, Chung et al., 16 Jul 2025).

Paper	Domain	Core object
(Yuge, 19 May 2026)	OT–IG geometry	Second-moment matrix $M_x$ and observable/unobservable cost decomposition
(Chung et al., 16 Jul 2025)	Explainable intrusion detection	Multi-granular Gaussian-based tokens for continuous features

The distinction is substantive. In the OT–IG formulation, the key objects are support coordinates $x \in \mathbb{R}^f$, Jacobians $J = \partial x/\partial \theta$, covariance matrices, Frobenius projections, and Fisher metrics. In the intrusion-detection formulation, the key objects are symbolic tokens, coherent normal and anomalous patterns, frequency-and-length weighted scores, and deterministic decision rules. A plausible implication is that “multi-granular” should be read contextually: in one case it concerns support-space geometry across scales or resolutions of $\omega$; in the other it concerns multiple Gaussian-based discretization precisions for feature values.

2. Support-space geometry and the OT–IG correspondence

In the geometric formulation, discretization begins with a continuous $f$-dimensional support with coordinates $x \in \mathbb{R}^f$ and a discretization obtained by translating a bounded convex set $\omega \subset \mathbb{R}^f$. Let $u$ denote the local coordinate in $\omega$ whose mean over $M_x$0 vanishes. The fundamental geometric object is the second-moment matrix

$M_x$1

where $M_x$2 denotes the volume of $M_x$3. Under this definition, $M_x$4 and $M_x$5 (Yuge, 19 May 2026).

When parameters $M_x$6 on the statistical manifold are directly identifiable with the support coordinates $M_x$7 in the precise sense that $M_x$8, the discretization cost at vanishing discretization scale $M_x$9 is characterized by the correspondence between OT and IG, yielding the exact formula

$x \in \mathbb{R}^f$0

Here, $x \in \mathbb{R}^f$1 denotes the Fisher metric associated with support-coordinate variations, $x \in \mathbb{R}^f$2 and $x \in \mathbb{R}^f$3 are nearby continuous distributions, and $x \in \mathbb{R}^f$4 is the probability density for infinitesimal parameter variation $x \in \mathbb{R}^f$5. The left-hand side $x \in \mathbb{R}^f$6 is the OT-measured discretization cost; the right-hand side is twice the expected KL divergence between nearby continuous distributions under $x \in \mathbb{R}^f$7 (Yuge, 19 May 2026).

This identifiable case directly ties the transport metric, through the $x \in \mathbb{R}^f$8-Wasserstein structure, to the Fisher metric on the statistical manifold. The formulation treats discretization-induced distortion as extrinsic to IG alone, but still expressible as a precise transport-information quantity once the parameter–support identifiability condition is satisfied. Multi-granular discretization enters here through the geometry of $x \in \mathbb{R}^f$9: varying the discretization cell across scales or resolutions modifies $J = \partial x/\partial \theta$0, and therefore modifies the total cost $J = \partial x/\partial \theta$1.

3. Rank-deficient parametrizations, Frobenius projection, and hidden transport sectors

The geometric framework is extended to general parametrizations $J = \partial x/\partial \theta$2 with local displacement map on the support space $J = \partial x/\partial \theta$3 and Jacobian $J = \partial x/\partial \theta$4. Given a covariance matrix $J = \partial x/\partial \theta$5 for parameter fluctuations, the induced covariance on the support is

$J = \partial x/\partial \theta$6

The observable covariances form the linear subspace

$J = \partial x/\partial \theta$7

If $J = \partial x/\partial \theta$8 is rank-deficient, then $J = \partial x/\partial \theta$9, meaning not all second-moment matrices on support space can be represented through parameter fluctuations. Consequently, the transport-information correspondence $\omega$0 cannot generally be reconstructed under such parametrizations, because certain physical displacements on the support are invisible to $\omega$1-variations (Yuge, 19 May 2026).

To address this, the framework introduces an orthogonal decomposition under the Frobenius inner product $\omega$2. One writes

$\omega$3

where $\omega$4 is the projection onto $\omega$5 and $\omega$6 is the orthogonal complement. Orthogonality is defined by

$\omega$7

This implies

$\omega$8

and therefore $\omega$9 and $f$0 are indistinguishable via quadratic forms induced by $f$1:

$f$2

The projection operator is defined through the Moore–Penrose pseudoinverse:

$f$3

and the Frobenius-orthogonal projection map $f$4 is

$f$5

The observable and unobservable components are thus

$f$6

with $f$7. Positive semidefiniteness is preserved: if $f$8, then $f$9.

Because discretization geometry is intrinsically a support-space property, the total transport cost is physically invariant to parametrization. Under rank-deficient parametrization, however, the cost admits the decomposition

$x \in \mathbb{R}^f$0

where $x \in \mathbb{R}^f$1 and $x \in \mathbb{R}^f$2 with a common proportional constant. The first term is reconstructable from $x \in \mathbb{R}^f$3-induced fluctuations, while the second is physically present but unobservable under the chosen parametrization. A common misconception is that rank deficiency destroys the total discretization cost; the framework instead states that rank deficiency breaks the standard IG interpretation of that cost, while preserving the invariant total $x \in \mathbb{R}^f$4 (Yuge, 19 May 2026).

4. Multi-resolution Gaussian tokenization for Interpretable Generalization

In the intrusion-detection formulation, IG begins with labeled flows $x \in \mathbb{R}^f$5 with binary labels $x \in \mathbb{R}^f$6. Each $x \in \mathbb{R}^f$7 is mapped into a set of symbolic attribute-value tokens. For categorical and ordinal features, the token is the literal category. Pattern discovery proceeds within each class by forming pairwise intersections of token sets: for Normal flows $x \in \mathbb{R}^f$8, the intersection of every pair $x \in \mathbb{R}^f$9 yields candidate normal patterns; for Anomalous flows $\omega \subset \mathbb{R}^f$0, the same process yields candidate anomalous patterns. A candidate pattern $\omega \subset \mathbb{R}^f$1 is coherent for class $\omega \subset \mathbb{R}^f$2 if and only if $\omega \subset \mathbb{R}^f$3 never appears as a subset in any instance of the opposite class $\omega \subset \mathbb{R}^f$4. The surviving sets are the Coherent Normal Patterns (CNP) and Coherent Anomalous Patterns (CAP), each stored with its absolute frequency $\omega \subset \mathbb{R}^f$5 and length $\omega \subset \mathbb{R}^f$6 (Chung et al., 16 Jul 2025).

IG-MD replaces the single discretization used for continuous features with multiple granularities. For each continuous feature $\omega \subset \mathbb{R}^f$7, the method estimates

$\omega \subset \mathbb{R}^f$8

then standardizes values by

$\omega \subset \mathbb{R}^f$9

Let $u$0 be the set of granularities, operationalized as a precision set $u$1. At granularity $u$2, with step size $u$3, the discretization mapping is

$u$4

Each discretized value is stored as a token that concatenates feature name or index, the granularity tag $u$5, and the rounded $u$6 value. Distinct granularities remain distinguishable, so the pattern miner can discover rules that mix resolutions (Chung et al., 16 Jul 2025).

The same construction can be expressed through uniform $u$7-interval bins of width $u$8, with boundaries

$u$9

The paper also records optional Gaussian-threshold variants based on $\omega$0 or Gaussian quantiles $\omega$1, while stating that the published implementation uses the $\omega$2-rounding formulation.

Once tokenization is complete, IG’s pairwise-intersection miner operates unchanged. At inference, an instance $\omega$3 is scored by

$\omega$4

In the multi-granular form,

$\omega$5

Labeling then follows three deterministic rules, in order: score dominance, a double-zero safeguard, and a statistical deviation guard-band based on $\omega$6. The framework preserves missing values as the literal token NaN, and it applies an anti-contradiction filter that removes any pair of instances whose entire symbolic representations are identical but carry opposite labels (Chung et al., 16 Jul 2025).

5. Computational procedure, complexity, and empirical behavior

The end-to-end IG-MD pipeline in intrusion detection consists of preprocessing, multi-granular tokenization, coherent pattern mining, guard-band estimation, and inference. Preprocessing relabels non-Normal classes as Anomalous, preserves missing values as NaN tokens, and estimates $\omega$7 for each continuous feature. Tokenization then emits $\omega$8 for each continuous feature and granularity, appends categorical and ordinal tokens as-is, and applies the anti-contradiction step. Pattern mining forms pairwise intersections separately in the Normal and Anomalous subsets, records pattern frequencies, and removes any pattern that violates coherence across classes. Inference rebuilds the token set for a test instance, computes $\omega$9 and $M_x$00, and applies the three deterministic rules (Chung et al., 16 Jul 2025).

The reported computational complexity is explicit. Let $M_x$01 be the number of samples; let $M_x$02 be the feature count; and let $M_x$03 be the number of granularities. Tokenization is $M_x$04 for numeric features, plus $M_x$05 for categorical features. With $M_x$06 and $M_x$07 denoting class sizes and $M_x$08 tokens per instance, pairwise intersections are $M_x$09. Coherence filtering depends on subset containment checks; inference is naively $M_x$10 subset checks and can be accelerated with token-to-pattern indexing (Chung et al., 16 Jul 2025).

The empirical evaluation covers NSL-KDD, UNSW-NB15, and UKM-IDS20. For the IG baseline, using only $M_x$11 training data, the reported performance is approximately $M_x$12 (Precision/Recall/AUC) on NSL-KDD, $M_x$13 on UNSW-NB15, and $M_x$14 on UKM-IDS20. Recall saturates at $M_x$15 with modest training fractions, such as $M_x$16–$M_x$17 on UNSW-NB15 and $M_x$18–$M_x$19 on UKM-IDS20 (Chung et al., 16 Jul 2025).

Replacing IG’s discretizer with multi-granular $M_x$20-rounding yields the reported UKM-IDS20 gains. Across all nine train–test splits, precision improves by $M_x$21 percentage points while recall remains approximately equal to $M_x$22. For $M_x$23 ($M_x$24 train), precision rises from $M_x$25 to $M_x$26; for $M_x$27, from $M_x$28 to $M_x$29. Average precision increases from $M_x$30 to $M_x$31. AUC stays in $M_x$32–$M_x$33. Error rate at $M_x$34 drops from $M_x$35 to $M_x$36 ($M_x$37 relative reduction), with one misclassification in $M_x$38 flows versus one in $M_x$39 previously. The paper describes the mechanism as a lightweight ensemble over granularities: the coarse layer rapidly flags deviations, while the fine layer refines ambiguous cases (Chung et al., 16 Jul 2025).

6. Interpretability, misconceptions, limitations, and prospective directions

Interpretability is central in the intrusion-detection formulation. Because bins are defined in standardized Gaussian units and the granularity tag is explicit in each token, rules remain auditable at the level of feature-value conditions. The paper gives an illustrative rule:

• IF duration_z at coarse granularity $M_x$40 is in $M_x$41, i.e., $M_x$42, AND
• bytes_out_z at fine granularity $M_x$43 is in $M_x$44, i.e., $M_x$45,
• THEN label Anomalous, with the rule lying in CAP and having $M_x$46 and $M_x$47 (Chung et al., 16 Jul 2025).

A worked discretization example uses a feature “packet_rate” with $M_x$48 packets/s, $M_x$49, and observed value $M_x$50. Then $M_x$51; at $M_x$52, $M_x$53 and the corresponding $M_x$54-bin is $M_x$55; at $M_x$56, $M_x$57 and the corresponding $M_x$58-bin is $M_x$59. This example makes explicit how the same value participates simultaneously in coarse and fine symbolic descriptions (Chung et al., 16 Jul 2025).

Two misconceptions are explicitly countered by the source materials. First, IG-MD in intrusion detection does not require features to be Gaussian; it uses $M_x$60 and $M_x$61 to standardize and define interpretable $M_x$62-based bins. Heavy tails or skew can be mitigated with robust estimators, winsorization, or monotone transforms. Second, in the OT–IG framework, rank-deficient parametrization does not erase the physical discretization cost; rather, it prevents full reconstruction of that cost as parameter-induced local indistinguishability, leaving an unobservable transport sector quantified by $M_x$63 (Chung et al., 16 Jul 2025, Yuge, 19 May 2026).

The limitations also differ by formulation. In the OT–IG setting, the framework assumes smooth mappings $M_x$64, existence of the Fisher metric $M_x$65 for support-coordinate variations, and the $M_x$66 discretization limit; the identifiability condition $M_x$67 is crucial for the full OT–IG correspondence. In the intrusion-detection setting, distribution shift and non-stationarity may require periodic re-estimation of $M_x$68 and $M_x$69; adding granularities increases tokens and candidate patterns; and extremely fine precisions on small datasets may overfit. The proposed future extensions likewise diverge: the OT–IG paper points to non-Euclidean supports, stochastic parametrizations, and adaptive multi-scale schemes where $M_x$70 varies to align with observable transport sectors defined by $M_x$71, while the intrusion-detection paper proposes adaptive granularity selection per feature, nonparametric discretization while preserving auditability, Gaussian mixtures for multimodal features, and automated hyperparameter tuning (Yuge, 19 May 2026, Chung et al., 16 Jul 2025).