Multivariate signal restoration via Multifold Graph Learning

• Channel relationships may differ across frequency components and observation windows, while noisy and incomplete observations make independent graph estimation unreliable.The challenge is to constrain graph estimation without losing component- and window-specific relationships while jointly restoring signals.Source: §1 · p. 1
• TVGMD [16], GraphDict, and IDGL address different parts of the problem but do not combine frequency-specific graphs with structure shared across modes and windows.TVGMD does not share graph structure across modes and windows; GraphDict does not decompose signals into frequency components; IDGL estimates one graph per window.Source: §1 · p. 1
• The method represents component- and window-specific graphs as nonnegative combinations of shared prototypes and jointly optimizes mode decomposition, graph learning, and signal restoration.Observations across components and windows contribute to estimating the shared prototypes.Source: §1 · p. 1
• The alternating optimization is unrolled into a self-supervised network with trainable regularization weights, step sizes, and filter coefficients.The network has only a few hundred trainable parameters.Source: §1 · p. 1
• The method achieves the lowest graph estimation error among compared methods on synthetic signals and competitive temperature completion with about 1/1500 as many parameters as a strong imputation baseline.On real-world temperature data, it ranks first or second across five missing-data settings.Source: §1 · p. 1

2. MULTIFOLD GRAPH MODEL

• Each observation window is modeled as a sum of K narrowband modes, each smooth on its own graph formed as a nonnegative mixture of R prototypes shared across modes and windows.Source: §2 · p. 1, Figure 1 · p. 2

2.1. Signal and Multifold Graph Model

• Each mode has a center frequency shared across windows and an associated channel graph; its edge weights are represented using shared prototypes with nonnegative mixing coefficients.Given masked observations, the modes, center frequencies, prototype graphs, and mixing coefficients are jointly estimated.Source: §2.1 · p. 2

2.2. Temporal Bandwidth and Graph Signal Smoothness

• Temporal bandwidth is defined using the one-sided DFT around a mode's center frequency, while graph smoothness measures weighted squared differences between adjacent-channel signals.The smoothness quantity is normalized by a fixed positive factor.Source: §2.2 · p. 2

3.1. Joint Problem and Block-Coordinate Updates

• The joint objective fits observed entries, keeps completed signals close to their mode sum, encourages narrowband modes and graph smoothness, and regularizes graph weights and node degrees.The prototypes have total edge weight N/2, fixing the scaling ambiguity; the numbers of modes and prototypes are fixed.Source: §3.1 · p. 2
• The nonconvex problem alternates signal reconstruction, frequency, mixing-coefficient, and prototype updates, repeating blocks until a stopping criterion or iteration limit is reached.The final graphs are used for signal and frequency updates before the modes are summed to reconstruct the signal.Source: §3.1 · pp. 2–3
• Mode updates apply a frequency-domain Wiener filter to each mode’s residual, while completed signals are updated with modes fixed and center frequencies use a weighted spectral centroid.Mixing coefficients are updated independently with modes and prototypes fixed; shared prototypes use contributions from all modes and windows, followed by rescaling that preserves each graph and the objective value.Source: §3.1 · p. 3

3.2. Unrolled Network

• The unrolled network has J layers corresponding to outer iterations, with fixed update counts and trainable regularization weights, relaxation exponents, and filter coefficients.A learnable polynomial surrogate replaces repeated graph-dependent linear-system inverses; normalized-Laplacian powers are evaluated through matrix–vector products.Source: §3.2 · p. 3

4. EXPERIMENTAL RESULTS

• Synthetic recovery uses 32 channels and 32 windows with mixed graph prototypes; weather completion tests five temperature masking patterns across 168 windows.Weather tasks include pointwise, station-window, and 6-, 24-, and 72-h gap masks; masked entries are excluded from preprocessing and fitting.Source: §4.1 · p. 3, §4.2 · p. 4

4.1. Synthetic Graph Recovery

• The R = 2 network achieves the lowest mean graph error across all ten synthetic conditions.Relative to R = 1, it reduces mean error by 13–17% for pointwise missing data and 7–18% for channel gaps.Source: §4.1 · p. 3, Figure 2 · p. 4

4.2. Weather Completion

• The proposed network ranks first or second in mean NMSE across five temperature-completion tasks and beats SAITS on all three gap tasks.It uses 488 trainable parameters, compared with approximately 730K for SAITS.Source: Table 1 · p. 4, §4.2 · p. 4
• Removing graph filtering raises NMSE in all five tasks, while independent graphs perform worse on station-window and all three gap tasks.The advantage of multiple prototypes is task-dependent: R = 1 performs better for 24-h gaps, whereas R = 3 performs better for 72-h gaps.Source: §4.2 · p. 4
• The free-frequency temperature variant recovers a mode within one DFT bin of the 24-h line in all 30 pointwise and station-window fits.The passage reports this recovery without prior knowledge of the period.Source: §4.2 · p. 4
• Across 145 stations, 12-h pressure amplitudes correlate with expected latitude dependence at r = 0.827 and directly fitted amplitudes at r = 0.945.Amplitudes are summarized by RMS over windows and averaged over three seeds.Source: §4.2 · p. 4

• The method jointly restores signals and learns component- and window-specific graphs as nonnegative combinations of shared prototypes.Source: §5 · p. 4
• Synthetic experiments show improved graph estimation accuracy over the compared methods.Source: §5 · p. 4
• On weather data, completion performance is competitive with substantially fewer trainable parameters than a self-attention-based model.Source: §5 · p. 4