Biorthogonal Graph Fourier Transform

• BGFT is a generalized spectral framework that uses pairs of biorthogonal eigenvectors to analyze non-normal and directed graph operators.
• It enables perfect reconstruction and the design of spectral filters even where classical orthogonal methods fail due to operator asymmetry.
• The framework provides robust sampling and reconstruction guarantees while addressing stability via condition number and biorthogonality measures.

The Biorthogonal Graph Fourier Transform (BGFT) is a generalized framework for spectral analysis of graph signals that extends classical orthogonal Graph Fourier Transform (GFT) methodologies to settings where the underlying graph operators—typically adjacency or Laplacian matrices—are non-symmetric or non-normal. BGFT leverages pairs of biorthogonal left and right eigenvectors to exactly diagonalize such operators, providing perfect reconstruction, canonical spectral filtering, and sampling/reconstruction guarantees in both undirected and directed networks, including regimes where traditional orthogonality and energy preservation collapse. This paradigm is central to recent advances in harmonic analysis on directed and non-normal graphs, as articulated in operator-centric frameworks using random-walk Laplacians, combinatorial Laplacians, or adjacency matrices (Gokavarapu, 25 Dec 2025, Gokavarapu, 1 Jan 2026, Gokavarapu, 13 Dec 2025, Gokavarapu, 15 Dec 2025, Pavez et al., 2020, Tremblay et al., 2015).

1. Algebraic Structure and Theoretical Foundations

BGFT arises from the failure of self-adjointness (symmetry) in operators representing directed networks. Given a graph $G=(V,E,w)$ with $n$ nodes, let $A\in\mathbb{R}^{n\times n}$ be its adjacency matrix and $$D_{\mathrm{out}}=\mathrm{diag}(d^{\mathrm{out}}_1,\dots,d^{\mathrm{out}}_n)$$ the out-degree matrix ($d^{\mathrm{out}}_i = \sum_j A_{ij}$). Common transforms include:

• Random-walk transition matrix: $P = D_{\mathrm{out}}^{-1}A$.
• Random-walk Laplacian: $L_{\mathrm{rw}} = I - P$.
• Combinatorial Laplacian: $L = D_{\mathrm{out}} - A$.

In undirected graphs or in reversible Markov regimes, these operators are (possibly weighted) self-adjoint and admit an orthonormal eigenbasis. For generic directed graphs, operators are non-self-adjoint and often non-normal ($LL^*\neq L^*L$), precluding an orthogonal eigenbasis and necessitating a biorthogonal paradigm (Gokavarapu, 25 Dec 2025, Gokavarapu, 1 Jan 2026).

Under diagonalizability, one constructs pairs of right eigenvectors $v_k$ and left eigenvectors $u_k$ satisfying $O v_k = \lambda_k v_k$, $u_k^* O = \lambda_k u_k^*$ (where $O$ represents $A$, $L$, or $P$ as appropriate), with mutual biorthogonality $u_j^* v_i = \delta_{ij}$. These sets furnish the BGFT coordinate system, synthesizing and analyzing signals in the spectral domain.

2. Biorthogonal Basis Construction and Transform Formulation

Construction of the BGFT is grounded in the biorthogonal eigenbasis:

• Spectral Decomposition: For a diagonalizable operator $O$,

$$O = V \Lambda V^{-1}, \quad \Lambda = \mathrm{diag}(\lambda_1,\ldots,\lambda_n),$$

where $V = [v_1 \ \cdots \ v_n]$ contains the right eigenvectors and $U^* = V^{-1}$, with columns $u_k$.

• Biorthogonality: $u_j^* v_i = \delta_{ij}.$
• Inner Product Adaptations: For reversible random walks or weighted Laplacians, BGFT can be formulated relative to a stationary distribution-weighted inner product: $\langle x, y \rangle_\pi = x^\top \Pi y$ with $\Pi = \mathrm{diag}(\pi)$ (Gokavarapu, 25 Dec 2025).

Forward (analysis) transform: For a signal $f \in \mathbb{C}^n$,

$$\hat f(k) = \begin{cases} u_k^* f, & \text{(standard BGFT)} \ (\psi_k^L)^\top \Pi f, & \text{(weighted, random-walk BGFT)} \end{cases}$$

Inverse (synthesis) transform:

$f = \sum_{k=1}^n \hat f(k) v_k$

or, in matrix language, $\hat f = U^* f$, $f = V \hat f$.

This formalism provides exact analysis and synthesis, independent of operator symmetry, and serves as the minimal spectral structure restoring invertibility absent orthogonality (Gokavarapu, 13 Dec 2025, Gokavarapu, 15 Dec 2025).

3. Spectral Ordering, Directed Frequencies, and Filter Design

For undirected (self-adjoint, normal) operators, spectral "frequency" corresponds either to eigenvalue magnitude (Laplacian: $|\lambda_k|$) or Rayleigh quotient. In BGFT for directed graphs:

• Diffusion-consistent ordering: For $P$ or $L_{\mathrm{rw}}$, define $\omega_k := \Re(1 - \lambda_k)$; modes with small $\omega_k$ decay slowly under diffusion (Gokavarapu, 25 Dec 2025).
• Adjacency-driven orderings: Alternative frequency notions may use $|\lambda_k|$ (damping amplitude) or $\arg \lambda_k$ (rotational/oscillatory ordering) (Gokavarapu, 13 Dec 2025).

Spectral filters are defined as $h(O)$ (e.g., polynomials, exponentials), yielding operator expressions $H = V\, h(\Lambda) \, V^{-1}$. Filtering acts diagonal in the BGFT domain: $\hat y = h(\Lambda)\hat f$ (Gokavarapu, 13 Dec 2025, Gokavarapu, 25 Dec 2025).

4. Energy Metrics, Parseval Bounds, and Conditioning

Unlike the orthogonal GFT, BGFT does not in general preserve energy in the spectral domain:

• Quadratic Form: $\|f\|_2^2 = \hat f^* G \hat f$ with Gram matrix $G = V^* V$.
• Two-sided norm bounds:

$$\sigma_{\min}^2(V) \| \hat f \|_2^2 \leq \| f \|_2^2 \leq \sigma_{\max}^2(V) \| \hat f \|_2^2$$

or equivalently $$1/\kappa(V) \|f\|_2 \leq \|\hat f\|_2 \leq \kappa(V)\|f\|_2$$, where $\kappa(V) = \|V\|_2\|V^{-1}\|_2$ (Gokavarapu, 1 Jan 2026, Gokavarapu, 15 Dec 2025).

The geometric distortion quantified by $\kappa(V)$ (the condition number of $V$) separates the norm equivalence regime ($\kappa(V)\approx1$ in normal or nearly symmetric graphs) from severe instability in highly non-normal cases.

5. Sampling, Reconstruction, and Stability

BGFT provides a rigorous theory for sampling and reconstruction on graphs, even when the underlying operator is non-normal:

• Bandlimited signals: For index set $\Omega\subset\{1,\ldots,n\}$ ($K=|\Omega|$) and $V_\Omega = [v_k]_{k\in\Omega}$,

$$f \in \text{span}\{v_k : k\in \Omega\} \implies f = V_\Omega c.$$

• Sampling: Let $M\subset V$ ($|M|=m$), define restriction $P_M\in \{0,1\}^{m\times n}$. If $P_M V_\Omega$ is full column rank, then in the noiseless case, $f$ is recovered exactly from $y = P_M f$ by

$$\hat c = (P_M V_\Omega)^{+} y, \quad \hat f = V_\Omega \hat c.$$

• Noise sensitivity: In the presence of noise $y = P_M f + \eta$,

$$\|\hat f - f\|_2 \leq \|V_\Omega\|_2 / \sigma_{\min}(P_M V_\Omega) \cdot \|\eta\|_2.$$

This cleanly distinguishes between instability due to non-orthogonality ($\|V_\Omega\|_2$) and sampling geometry ($1/\sigma_{\min}$) (Gokavarapu, 13 Dec 2025, Gokavarapu, 1 Jan 2026).

• Filter/reconstruction conditioning: Error amplification by up to $\kappa(V)$ is fundamentally tied to operator non-normality, not mere asymmetry (Gokavarapu, 1 Jan 2026, Gokavarapu, 15 Dec 2025, Gokavarapu, 25 Dec 2025).

6. Two-Channel/Band BGFT and Multiresolution Schemes

Biorthogonal GFTs enable perfect-reconstruction, critically-sampled, two-channel filterbanks generalizing classical subband decomposition to arbitrary and directed graphs.

• For undirected graphs with positive semi-definite $L$ and $Q\succ0$, the BGFT basis solves $L u_i = \lambda_i Q u_i$ with $Q$-orthonormality $u_i^\top Q u_j = \delta_{ij}$ (Pavez et al., 2020).
• A spectral folding property allows construction of two-channel subband decompositions: for any partition, the spectrum folds about the midpoint (e.g., $1$ for normalized Laplacian).
• Analysis/synthesis filters are constructed as $U \, \mathrm{diag}(h_j(\lambda_i)) \, U^\top Q$, with biorthogonality and perfect-reconstruction equations for filter design.
• Critically-sampled, compactly-supported, biorthogonal BGFT filterbanks also arise via subgraph-based approaches using local Fourier bases on subgraphs, enabling locality and algorithmic scalability (Tremblay et al., 2015).

7. Empirical Characterizations and Application Domains

BGFTs yield qualitative and quantitative distinctions between "normal" directed graphs (directed cycles, reversible Markov chains) and genuinely non-normal architectures (perturbed cycles, general digraphs):

Graph/Operator	Symmetry	Normality	$\kappa(V)$	BGFT Energy Distortion	Stability
Undirected/Laplacian	symmetric	normal	$1$	none (Parseval identity)	stable
Directed cycle	non-symmetric	normal	$1$	none	stable
Perturbed directed cycle	non-symmetric	non-normal	$\gg 1$	possibly severe	instability, amplification

Simulations confirm that even minor departures from normality dramatically degrade filter/reconstruction robustness, correlating with $\kappa(V)$ and Henrici's departure-from-normality $\Delta(L)$ (Gokavarapu, 1 Jan 2026, Gokavarapu, 15 Dec 2025, Gokavarapu, 25 Dec 2025). For large-scale, high-dimensional graph datasets (e.g., 3D point clouds), multiresolution BGFTs can outperform bipartite or strictly orthogonal filterbanks in rate-distortion and run-time, especially for non-bipartite structures (Pavez et al., 2020, Tremblay et al., 2015).

8. Outlook and Generalizations

BGFT subsumes classical orthogonal GFT in regimes of symmetry or operator normality; in the general (non-normal, asymmetric) case, it quantifies all analytic and operational departures—energy distortion, noise sensitivity, stability—through explicit metrics ($\kappa(V)$, spectral geometry, sampling set stability). The framework admits generalizations to defective operators (Schur/Jordan decompositions), regularized designs to control conditioning, and applications to direction-aware filtering, sampling design, denoising, and beyond (Gokavarapu, 1 Jan 2026, Gokavarapu, 25 Dec 2025).

Ongoing research addresses adaptive sampling, regularized transforms for highly ill-conditioned spectra, extensions to complex- or magnetic Laplacians, and efficient polynomial BGFT constructions for large graphs. In all cases, BGFT provides the unique algebraic language restoring invertibility and spectral diagonalization in general directed network settings.