Spectral Redundancy Index

• Spectral Redundancy Index is a quantitative measure defining repetition and compressibility in various spectral domains, including hyperspectral imaging and graph spectra.
• It enables the decoupling of redundancy from relevance, using methods like symmetric uncertainty and dimensionality estimates to enhance data interpretation.
• The index guides practical applications from parameter-efficient fine-tuning in LoRA to mitigating over-smoothing in graph filters, highlighting its cross-domain significance.

Searching arXiv for relevant papers on spectral redundancy index and related uses. Spectral Redundancy Index denotes a family of quantitative notions used to measure repetition, compressibility, or non-distinctness in spectral information. In the literature, the term is not standardized to a single formula. In hyperspectral imaging, it can denote pairwise redundancy between spectral bands or the gap between ambient band count and intrinsic spectral subspace dimension; in spectral dimensionality reduction, it is naturally tied to the predictability of an embedding coordinate from previous ones; in parameter-efficient fine-tuning, it quantifies unused spectral degrees of freedom in LoRA updates; and in spectral graph theory, it is an explicit ratio comparing the number of non-isomorphic connected induced subgraphs with the number of distinct induced-subgraph spectral radii (Sarhrouni et al., 2012, Rasti et al., 2016, Blau et al., 2016, Cheng et al., 20 Jun 2025, Kumar et al., 2024, Kumar et al., 14 Jul 2025).

1. Terminological scope

Taken together, these works suggest that “spectral” may refer to spectral bands, singular spectra, graph Laplacian eigenmodes, or adjacency spectral radii, and that “redundancy” may mean shared information, compressible coordinates, repeated eigen-directions, or repeated spectral radii across induced subgraphs (Sarhrouni et al., 2012, Blau et al., 2016, Bison et al., 17 Jul 2025, Kumar et al., 14 Jul 2025).

Domain	Quantity used as redundancy measure	Interpretation
Hyperspectral band selection	$U(B_i,B_j)=2\frac{I(B_i;B_j)}{H(B_i)+H(B_j)}$	Pairwise band redundancy
Hyperspectral subspace ID	$1-\hat r/p$	Redundancy as excess ambient bands over intrinsic spectral dimension
Spectral embeddings	Predictability of $f_i$ from $f_1,\dots,f_{i-1}$	Repeated eigen-directions
LoRA / SeLoRA	$1-\frac{N_{\text{eff}}}{N}$ or sparsity-based variants	Unused spectral degrees of freedom
Graph reservoirs	Concentration of filtered spectral energy	Over-smoothing and redundant random walks
Spectral graph theory	$r(G)=\frac{b(G)}{c(G)}$	Repeated spectral radii among connected induced subgraphs

A recurring misconception is that a Spectral Redundancy Index must be a single scalar attached to an entire dataset or model. The literature does not support that restriction. In some settings it is pairwise and local, as with band-to-band Symmetric Uncertainty or cosine similarity; in others it is global, as with $1-\hat r/p$ or $r(G)=b(G)/c(G)$ (Sarhrouni et al., 2012, Lin et al., 2024, Rasti et al., 2016, Kumar et al., 14 Jul 2025).

2. Hyperspectral imaging: band-level redundancy

In hyperspectral classification, redundancy is treated separately from relevance. “Application of Symmetric Uncertainty and Mutual Information to Dimensionality Reduction and Classification of Hyperspectral Images” defines relevance as the information a band carries about class labels and redundancy as shared information between bands, irrespective of class labels (Sarhrouni et al., 2012). Mutual information is

$$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$

and Symmetric Uncertainty is

$U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$

with $1-\hat r/p$0. The paper explicitly interprets SU as “how much information is shared between $1-\hat r/p$1 and $1-\hat r/p$2 relatively to all information contained in both $1-\hat r/p$3 and $1-\hat r/p$4.”

At the pairwise band level, the paper supports the direct interpretation

$1-\hat r/p$5

as a spectral redundancy index between bands (Sarhrouni et al., 2012). High values indicate near-duplicate bands; low values indicate near-independence. The synthetic examples make the interpretation explicit: $1-\hat r/p$6 is described as “practically the same,” whereas $1-\hat r/p$7 is “practically disjoint.”

The associated filter algorithm has two phases. First, it thresholds $1-\hat r/p$8 to retain relevant bands. Second, it suppresses redundancy by computing pairwise SU on the surviving bands and adding a candidate band to the selected subset only if its Symmetric Uncertainty with all already selected bands is below a redundancy threshold. The two thresholds, $1-\hat r/p$9 and $f_i$0, decouple closeness to the ground truth from similarity among bands (Sarhrouni et al., 2012). On AVIRIS 92AV3C, with redundancy threshold $f_i$1 and relevance threshold $f_i$2, the method selected 42 bands and obtained 84.16% classification accuracy; the paper also reports an “interesting” regime with about 40 bands and about 80% accuracy, and a “very important zone” with 80% accuracy using only 19 bands.

The same domain later reappears in masked autoencoding. “Boosting Spatial-Spectral Masked Auto-Encoder Through Mining Redundant Spectra for HSI-SAR/LiDAR Classification” characterizes redundant bands as highly similar spectra that create information leakage during reconstruction (Lin et al., 2024). Given a randomly chosen base band $f_i$3, the similarity to band $f_i$4 is

$f_i$5

and MRS masks the top-$f_i$6 most similar bands: $f_i$7 Here cosine similarity functions as an implicit pairwise spectral redundancy index. The operational meaning is straightforward: higher $f_i$8 means greater redundancy with respect to the anchor band, so those bands are masked together to prevent shortcut reconstruction (Lin et al., 2024).

A plausible implication is that hyperspectral SRI is best viewed at two scales. The first is pairwise, where SU or cosine similarity ranks individual band pairs. The second is subset-level, where small internal redundancy is enforced by thresholding or by jointly masking highly similar groups. The former is explicit in both papers; the latter follows directly from their decision rules (Sarhrouni et al., 2012, Lin et al., 2024).

3. Intrinsic dimensionality and non-redundant spectral representations

A second major use of spectral redundancy concerns intrinsic dimensionality. “Hyperspectral Subspace Identification Using SURE” models the observed cube as

$f_i$9

where $f_1,\dots,f_{i-1}$0 is the spectral rank or subspace dimension (Rasti et al., 2016). Spectral redundancy here means that the true signal lies in a subspace of much lower dimension than the original $f_1,\dots,f_{i-1}$1-band space. After whitening, HySURE estimates $f_1,\dots,f_{i-1}$2 and the sparsity parameter $f_1,\dots,f_{i-1}$3 by minimizing

$f_1,\dots,f_{i-1}$4

with

$f_1,\dots,f_{i-1}$5

The paper itself does not name a Spectral Redundancy Index, but it provides a direct dimensionality-based construction: $f_1,\dots,f_{i-1}$6 This definition is technically consistent with the HySURE framework because $f_1,\dots,f_{i-1}$7 is the SURE-selected intrinsic spectral dimension (Rasti et al., 2016). The reported examples are strongly redundant in this sense: on a simulated $f_1,\dots,f_{i-1}$8-band dataset at SNR $f_1,\dots,f_{i-1}$9 dB, HySURE selected $1-\frac{N_{\text{eff}}}{N}$0, giving $1-\frac{N_{\text{eff}}}{N}$1; on Indian Pines with 220 bands it selected $1-\frac{N_{\text{eff}}}{N}$2 or $1-\frac{N_{\text{eff}}}{N}$3, giving about $1-\frac{N_{\text{eff}}}{N}$4; and on Cuprite with 206 bands it selected $1-\frac{N_{\text{eff}}}{N}$5, giving about $1-\frac{N_{\text{eff}}}{N}$6. These values quantify redundancy as the fraction of ambient spectral dimensions not needed by the estimated signal subspace.

A more refined variant is also suggested by the same paper. Since $1-\frac{N_{\text{eff}}}{N}$7 counts active coefficients after soft-thresholding, one can define a per-location effective spectral dimension

$1-\frac{N_{\text{eff}}}{N}$8

and then

$1-\frac{N_{\text{eff}}}{N}$9

This distinguishes redundancy of the original band space from additional redundancy within the chosen spectral subspace itself (Rasti et al., 2016).

The same shift from ambient dimensionality to non-redundant coordinates appears in “Non-Redundant Spectral Dimensionality Reduction” by Blau and Michaeli (Blau et al., 2016). Their central observation is the “repeated Eigen-directions” phenomenon: orthogonal embedding coordinates can still be redundant because one coordinate may be a nonlinear function of previous ones. They define a sequence $r(G)=\frac{b(G)}{c(G)}$0 as non-redundant if no $r(G)=\frac{b(G)}{c(G)}$1 can be expressed as a function of the preceding coordinates, and replace orthogonality by the unpredictability constraint

$r(G)=\frac{b(G)}{c(G)}$2

They prove that this constraint prevents redundancy.

A natural per-coordinate SRI suggested by their framework is a predictability coefficient obtained by regressing $r(G)=\frac{b(G)}{c(G)}$3 on previous coordinates: $r(G)=\frac{b(G)}{c(G)}$4 High values indicate that the coordinate is largely predictable from previous ones and is therefore redundant; low values indicate non-redundancy. The paper’s normalized regression error plays exactly this diagnostic role in the MNIST experiments, where higher normalized error is described as “less redundant” (Blau et al., 2016).

4. Spectral redundancy in LoRA and spectral encoding

In parameter-efficient fine-tuning, spectral redundancy refers neither to spectral bands nor to graph spectra, but to unused degrees of freedom in low-rank updates. “Revisiting LoRA through the Lens of Parameter Redundancy: Spectral Encoding Helps” distinguishes rank redundancy from density redundancy in standard LoRA

$r(G)=\frac{b(G)}{c(G)}$5

with $r(G)=\frac{b(G)}{c(G)}$6 and $r(G)=\frac{b(G)}{c(G)}$7 (Cheng et al., 20 Jun 2025). The paper’s empirical finding is that reducing density redundancy does not degrade expressiveness, whereas reducing rank can hurt performance substantially.

SeLoRA reparameterizes LoRA in a sparse spectral domain. It introduces sparse spectral matrices $r(G)=\frac{b(G)}{c(G)}$8 and $r(G)=\frac{b(G)}{c(G)}$9, and sets

$1-\hat r/p$0

where $1-\hat r/p$1 is an inverse Fourier or wavelet transform (Cheng et al., 20 Jun 2025). If $1-\hat r/p$2 is the sparse ratio, the trainable parameter count becomes

$1-\hat r/p$3

instead of $1-\hat r/p$4. The paper does not explicitly define a quantity called Spectral Redundancy Index, but it repeatedly frames the problem as the gap between full spectral capacity and the smaller number of spectral coefficients actually needed.

The most direct formulation proposed in the synthesis of the paper is

$1-\hat r/p$5

where $1-\hat r/p$6 is the total number of spectral coefficients and $1-\hat r/p$7 is the minimal number needed to achieve a target approximation quality or validation performance (Cheng et al., 20 Jun 2025). Closely related variants are

$1-\hat r/p$8

for the architectural sparsity ratio itself, and

$1-\hat r/p$9

for redundancy measured through the singular spectrum of $r(G)=b(G)/c(G)$0.

Two auxiliary spectral diagnostics further clarify the meaning of redundancy in this setting. If

$r(G)=b(G)/c(G)$1

the paper considers the Amplification Factor

$r(G)=b(G)/c(G)$2

and the Reverse Amplification Factor

$r(G)=b(G)/c(G)$3

Empirically, SeLoRA has lower RAF than LoRA, which the paper interprets as amplifying useful new directions while suppressing already-emphasized ones (Cheng et al., 20 Jun 2025). This supports a subspace-overlap reading of spectral redundancy: redundant updates are those that spend capacity on already saturated spectral directions.

5. Graph filters, random walks, and spectral concentration

“A Spectral Interpretation of Redundancy in a Graph Reservoir” treats redundancy as a property of graph filtering and repeated random-walk propagation rather than of induced-subgraph spectral radii (Bison et al., 17 Jul 2025). For an undirected graph, the symmetric normalized Laplacian is

$r(G)=b(G)/c(G)$4

with eigen-decomposition

$r(G)=b(G)/c(G)$5

Repeated propagation acts as a low-pass filter and can induce over-smoothing, meaning that graph signals collapse toward low-frequency eigenmodes.

The paper’s Fairing-based reservoir implements the filter

$r(G)=b(G)/c(G)$6

whose frequency response is

$r(G)=b(G)/c(G)$7

Redundancy then has two linked meanings: excessive concentration of energy in a small low-frequency subspace, and excessive contribution from redundant random walks, especially tottering walks and vacuous-step multiplicities (Bison et al., 17 Jul 2025).

The paper does not introduce a scalar SRI by name, but it reconstructs one naturally from the filter. One proposed option is an entropy-based concentration index

$r(G)=b(G)/c(G)$8

with normalized spectral weights

$r(G)=b(G)/c(G)$9

Another is a low-frequency dominance ratio

$$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$0

Both quantify how strongly the reservoir collapses the representation onto a narrow spectral band (Bison et al., 17 Jul 2025).

The random-walk derivation makes the redundancy interpretation explicit. Since

$$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$1

powers of the transition matrix $$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$2 aggregate paths of multiple lengths. The paper identifies redundancy from tottering walks and from multiple vacuous-step encodings of the same effective path. Tuning $$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$3 and $$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$4 is therefore interpreted as modulating the contribution of redundant random walks rather than merely adding or removing smoothing (Bison et al., 17 Jul 2025).

6. Spectral graph theory: repeated induced-subgraph spectral radii

The most explicit formal definition of Spectral Redundancy Index occurs in spectral graph theory. Following Seeger, the complementarity spectrum of a connected graph $$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$5 is

$$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$6

where $$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$7 is the set of connected induced subgraphs and $$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$8 is the spectral radius of $$I(A,B)=\sum p(A,B)\,\log_2\frac{p(A,B)}{p(A)\,p(B)},$$9 (Kumar et al., 2024, Kumar et al., 14 Jul 2025). If $U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$0 is the number of non-isomorphic connected induced subgraphs and $U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$1 is the number of distinct spectral radii among them, then the spectral redundancy of $U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$2 is

$U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$3

For a graph family $U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$4, the spectral redundancy index is

$U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$5

For pineapple graphs $U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$6, “On the spectral redundancy of pineapple graphs” gives a complete hereditary induced-subgraph description and proves

$U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$7

(Kumar et al., 2024). For $U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$8 with $U(A,B)=2 \cdot \frac{MI(A,B)}{H(A)+H(B)},$9,

$1-\hat r/p$00

and for fixed $1-\hat r/p$01 or fixed $1-\hat r/p$02,

$1-\hat r/p$03

Thus redundancy becomes asymptotically negligible in large pineapple graphs.

“On connected graphs with finite spectral redundancy index and Pythagorean triplets” develops the same concept for garlic graphs $1-\hat r/p$04 (Kumar et al., 14 Jul 2025). It proves

$1-\hat r/p$05

and

$1-\hat r/p$06

Equal spectral radii among non-isomorphic garlic graphs are characterized by algebraic relations in $1-\hat r/p$07, and Pythagorean triples enter through the condition

$1-\hat r/p$08

A Pythagorean triple can generate four non-isomorphic garlic graphs, together with a star, sharing the same spectral radius. Despite these infinite coincidence constructions, the family-level index is finite: $1-\hat r/p$09 This is the sharp spectral redundancy index for the family of garlic graphs (Kumar et al., 14 Jul 2025).

This graph-theoretic meaning is exact and canonical, but it is domain-specific. It should not be conflated with the information-theoretic, subspace-dimensional, or filter-concentration meanings discussed earlier. The literature instead supports a broader conclusion: Spectral Redundancy Index is a unifying label for formally different quantities that all compare nominal spectral complexity with effectively distinct spectral content (Sarhrouni et al., 2012, Rasti et al., 2016, Cheng et al., 20 Jun 2025, Bison et al., 17 Jul 2025, Kumar et al., 14 Jul 2025).