Sparsity-Promoting Dynamic Mode Decomposition

• Sparsity-Promoting Dynamic Mode Decomposition is a method that imposes an ℓ1 penalty on modal amplitudes to select a parsimonious subset of DMD modes.
• It uses a convex optimization framework solved by ADMM to balance reconstruction fidelity against model complexity in global snapshot data.
• The approach has been applied to turbulent flows, plasma diagnostics, and climate studies, yielding compact, interpretable models from dense modal libraries.

Searching arXiv for recent and foundational papers on Sparsity-Promoting Dynamic Mode Decomposition. Sparsity-Promoting Dynamic Mode Decomposition is a variant of Dynamic Mode Decomposition (DMD) that selects a parsimonious subset of DMD modes by imposing sparsity on the vector of modal amplitudes while preserving good reconstruction of the snapshot sequence. In its canonical form, introduced by Jovanović, Schmid, and Nichols, the method leaves the DMD eigenvalues and spatial modes unchanged and instead solves a convex optimization problem that balances least-squares reconstruction accuracy against an $\ell_1$ penalty on amplitudes (Jovanović et al., 2013). This formulation addresses a central limitation of standard DMD: the full DMD expansion is often too large for interpretation or reduced-order modeling, and naive ranking by amplitude can be misleading because DMD modes are generally non-orthogonal (Jovanović et al., 2013). Subsequent work has applied or adapted the idea in settings ranging from turbulent wind-farm flows and plasma diagnostics to climate, traffic, and short-term weather analysis, while also motivating adjacent variants in compressed, projected, multiresolution, and transient settings (Dai et al., 2022, Kaptanoglu et al., 2019, Zhang et al., 8 Jul 2025, Zhang et al., 17 Jun 2025).

1. Canonical formulation

The standard DMD setting begins with equispaced snapshots

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$

assembled into

$$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$

with the DMD ansatz

$\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$

Using the economy-size singular value decomposition

$\Psi_0 = U \Sigma V^*,$

the reduced operator is

$F = U^*\Psi_1 V \Sigma^{-1}.$

If

$$F = Y D_\mu Z^*, \qquad D_\mu = \operatorname{diag}(\mu_1,\ldots,\mu_r),$$

then the DMD modes are

$\phi_i = U y_i, \qquad \Phi = UY,$

and the snapshots are approximated by

$$\psi_t \approx \sum_{i=1}^r \phi_i \mu_i^t \alpha_i.$$

Collecting all snapshots yields

$\Psi_0 \approx \Phi D_\alpha V_{\!and},$

where

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$0

and

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$1

The amplitudes are therefore determined by the global least-squares problem

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$2

This global reconstruction objective, rather than a fit to a single initial condition alone, is the basis of canonical Sparsity-Promoting DMD (Jovanović et al., 2013).

The least-squares term can be written as

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$3

with

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$4

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$5

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$6

Without sparsity regularization, the optimal amplitudes are

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$7

The canonical sparsity-promoting problem replaces this dense estimate by

$$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$8

where $$\{\psi_0,\psi_1,\ldots,\psi_N\}, \qquad \psi_t \in \mathbb{C}^M,$$9 controls the tradeoff between approximation quality and the number of active modes (Jovanović et al., 2013).

2. Optimization structure and support refinement

The ideal sparse problem would penalize the number of nonzero amplitudes directly through a cardinality term. In the canonical paper, this intractable objective is relaxed to an $$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$0-regularized convex optimization problem, which admits a globally optimal solution and is solved by the alternating direction method of multipliers (ADMM) (Jovanović et al., 2013).

Introducing an auxiliary variable $$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$1 gives

$$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$2

with

$$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$3

The augmented Lagrangian is

$$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$4

ADMM alternates between an $$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$5-update, a $$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$6-update, and a dual update,

$$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$7

$$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$8

$$\Psi_0 = \begin{bmatrix} \psi_0 & \psi_1 & \cdots & \psi_{N-1} \end{bmatrix}, \qquad \Psi_1 = \begin{bmatrix} \psi_1 & \psi_2 & \cdots & \psi_N \end{bmatrix},$$9

with the explicit quadratic solve

$\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$0

and a soft-thresholding step for $\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$1 (Jovanović et al., 2013).

A defining feature of the canonical method is the post-selection “polishing” step. After the sparse solve identifies which entries should be zero, the nonzero amplitudes are recomputed by solving

$\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$2

where $\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$3 encodes the zero pattern. This constrained least-squares refinement removes the shrinkage bias introduced by the $\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$4 penalty and yields the final sparse amplitude vector (Jovanović et al., 2013). Later application papers in wind-farm flow analysis and plasma diagnostics follow the same conceptual pattern: a global sparse reconstruction objective, $\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$5 regularization on amplitudes, ADMM solution, and support refinement (Dai et al., 2022, Kaptanoglu et al., 2019).

3. Interpretation of sparsity and modal relevance

Sparsity-Promoting DMD does not change the extracted DMD eigenvalues or spatial modes. Its action is entirely on the amplitude vector. The method therefore answers a specific question: given a DMD basis already computed from the data, which subset of modes best reconstructs the full snapshot sequence at a prescribed model complexity (Jovanović et al., 2013)?

This emphasis distinguishes it from amplitude ranking heuristics. In the canonical paper, large amplitudes alone are not reliable indicators of importance because DMD modes are generally non-orthogonal, and a strongly damped mode can have a large coefficient while contributing only briefly to the full trajectory (Jovanović et al., 2013). The wind-farm study makes the same point in application-specific terms: amplitude-prioritized selection tends to favor small-scale, high-frequency, rapidly decaying modes, whereas sparsity-promoting selection favors large coherent structures with low frequency and small decay rates that matter to the global reconstruction (Dai et al., 2022). The plasma study similarly reports that optimized DMD gives the highest prediction accuracy, but sparsity-promoting DMD yields physically interpretable models that avoid overfitting and isolate the major magnetic modes (Kaptanoglu et al., 2019).

A common misconception is that Sparsity-Promoting DMD is a different spectral decomposition. It is not. The canonical formulation retains the DMD spectrum and modal basis and performs sparse amplitude selection on top of that basis (Jovanović et al., 2013). Another misconception is that the method necessarily keeps the largest-amplitude modes. The optimization is global over the full time series, so the retained support can differ substantially from a simple ranking by $\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$6 (Dai et al., 2022, Jovanović et al., 2013).

This suggests a useful editorial distinction. Canonical SP-DMD performs global support selection: a mode is either retained for reconstruction of the whole sampled interval or discarded. Several later papers depart from this by using sparse coding against a single frame, sparse projections, or time-varying amplitudes. Those variants are related, but they are not equivalent to the original formulation (Erichson et al., 2015, Murshed et al., 2020, Tanaka et al., 14 Aug 2025).

4. Relationship to adjacent DMD variants

Several later developments are closely related to SP-DMD but differ in what is made sparse.

The paper "Compressed Dynamic Mode Decomposition for Background Modeling" formulates sparse mode selection for video background modeling as

$\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$7

and solves it with Orthogonal Matching Pursuit (OMP), optionally in compressed coordinates (Erichson et al., 2015). It is explicitly motivated by Jovanović et al., but it does not solve the canonical global $\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$8-regularized amplitude problem over all snapshots. Its main novelty is computational scalability through compressed DMD, with sparsity used as a practical background-mode selection heuristic (Erichson et al., 2015).

The paper "Projection assisted Dynamic Mode Decomposition of large scale data" uses sparse random projection matrices in the preprocessing stage,

$\psi_{t+1} = A \psi_t, \qquad \Psi_1 = A \Psi_0.$9

but it does not impose sparsity on mode amplitudes and therefore is not SP-DMD in the canonical sense (Murshed et al., 2020).

The traffic paper "Anti-circulant dynamic mode decomposition with sparsity-promoting for highway traffic dynamics analysis" returns to canonical amplitude sparsification after introducing an anti-circulant delay embedding tailored to stable periodic behavior. Its sparse step is

$\Psi_0 = U \Sigma V^*,$0

solved using ADMM and followed by support refinement (Wang et al., 2023). Here the novelty lies in the anti-circulant embedding, while the sparsity mechanism is recognizably Jovanović-style amplitude selection (Wang et al., 2023).

The 2025 paper on transient activity introduces time-varying amplitudes,

$\Psi_0 = U \Sigma V^*,$1

with sparsity and smoothness regularization across time. Its central departure from canonical SP-DMD is that sparsity is imposed at each time step rather than on a single global amplitude vector (Tanaka et al., 14 Aug 2025). A plausible implication is that this extension addresses transient support changes that the original static-amplitude model cannot represent.

The paper "Sparse-mode Dynamic Mode Decomposition for Disambiguating Local and Global Structures" shifts sparsity from amplitudes to the entries of the amplitude-scaled mode matrix,

$\Psi_0 = U \Sigma V^*,$2

to localize the spatial support of modes themselves (Ichinaga et al., 26 Jul 2025). This is a distinct notion of sparsity from canonical SP-DMD, whose sparsity is in the mode index set rather than the mode entries.

Finally, "Parsimonious Dynamic Mode Decomposition" positions itself as an automated alternative to SP-DMD. It replaces the $\Psi_0 = U \Sigma V^*,$3-regularization path over $\Psi_0 = U \Sigma V^*,$4 with OMP-based greedy selection and an automatic stopping rule. Its contribution is comparative: SP-DMD remains the benchmark, but parsDMD seeks to remove manual sparsity-parameter tuning (Das et al., 2024).

5. Applications and empirical behavior

The canonical SP-DMD paper demonstrates the method on plane Poiseuille flow, a screeching supersonic jet, and experimental cylinder-bundle flow (Jovanović et al., 2013). In the Poiseuille example, reducing from 26 DMD modes to 13 increases least-squares residual by only about $\Psi_0 = U \Sigma V^*,$5, while further reduction reveals a physically interpretable subset consisting of one fast-branch mode, one slow-branch mode, and the unstable Tollmien–Schlichting mode (Jovanović et al., 2013). In the jet example, a sparse model with three nonzero modes retains the mean flow and a conjugate pair representing the dominant screech tone, recovering a Strouhal number of

$\Psi_0 = U \Sigma V^*,$6

in good agreement with the measured screech frequency (Jovanović et al., 2013). In the cylinder-bundle experiment, the sparse three-mode model isolates the dominant low-frequency swaying behavior and recovers a frequency of $\Psi_0 = U \Sigma V^*,$7, close to the measured $\Psi_0 = U \Sigma V^*,$8 (Jovanović et al., 2013).

Later application papers reinforce the same practical value. In wind-farm LES, SP-DMD selects low-frequency, weakly decaying, large-scale coherent structures and yields lower performance loss than amplitude-prioritized truncation when only a limited number of modes is retained (Dai et al., 2022). The plasma study reports that SP-DMD isolates three dominant large-scale magnetic structures in experiment, supports discovery of a previously unobserved rotating $\Psi_0 = U \Sigma V^*,$9 structure in simulation, and remains interpretable even with sparse sensor coverage (Kaptanoglu et al., 2019). In short-term weather simulations, a Koopman/SPDMD workflow uses sparse amplitudes to extract transient growing and decaying modes associated with warm bubble-like convective patterns, with a clear tradeoff between mode count and reconstruction loss (Zhang et al., 17 Jun 2025). In long-term Pacific sea-surface temperature analysis, SPDMD reduces monthly SST dynamics from 600 modes to about 33 dominant ones and seasonal dynamics from over 500 modes to as few as 19 or 9 at only a few percent performance loss, emphasizing annual and quasi-periodic climate components (Zhang et al., 8 Jul 2025).

The following table summarizes representative uses of canonical or near-canonical sparse amplitude selection.

Setting	Sparse object	Solver / mechanism
Canonical SP-DMD (Jovanović et al., 2013)	Global DMD amplitudes $F = U^*\Psi_1 V \Sigma^{-1}.$0	$F = U^*\Psi_1 V \Sigma^{-1}.$1-regularized convex optimization with ADMM
Wind-farm LES (Dai et al., 2022)	Global DMD amplitudes $F = U^*\Psi_1 V \Sigma^{-1}.$2	SP-DMD following Jovanović et al., solved by ADMM
Plasma diagnostics (Kaptanoglu et al., 2019)	Global DMD amplitudes $F = U^*\Psi_1 V \Sigma^{-1}.$3	Jovanović-style $F = U^*\Psi_1 V \Sigma^{-1}.$4 regularization with ADMM
Traffic circDMDsp (Wang et al., 2023)	Global amplitudes $F = U^*\Psi_1 V \Sigma^{-1}.$5 after anti-circulant embedding	ADMM + equality-constrained quadratic program
Weather Koopman/SPDMD (Zhang et al., 17 Jun 2025)	Global amplitudes $F = U^*\Psi_1 V \Sigma^{-1}.$6	ADMM + polishing step

These results consistently show that sparse amplitude selection is most valuable when the full DMD spectrum is too dense for interpretation and when the dynamics of interest are represented by a small number of coherent structures embedded in a larger modal library (Jovanović et al., 2013, Dai et al., 2022, Kaptanoglu et al., 2019).

6. Practical limitations and broader significance

The main limitation of canonical SP-DMD is that its quality depends on the underlying DMD basis. If standard DMD produces poor eigenvalues or modes, sparsity promotion cannot repair the spectral model; it can only select among the modes already available (Jovanović et al., 2013). A second limitation is the need to choose the regularization parameter $F = U^*\Psi_1 V \Sigma^{-1}.$7. The original method explores a sparsity path and compares residual loss against cardinality, but it does not provide a universal automatic rule for selecting $F = U^*\Psi_1 V \Sigma^{-1}.$8 (Jovanović et al., 2013). Later comparative work treats this as a practical weakness and proposes alternatives such as LARS-based or OMP-based mode selection to avoid manual tuning (Graff et al., 2019, Das et al., 2024).

A further limitation emerges in strongly transient or nonstationary problems. Because canonical SP-DMD uses a single global amplitude vector, it cannot represent support that changes over time. This motivates extensions with time-varying sparse amplitudes and smoothness penalties (Tanaka et al., 14 Aug 2025). Likewise, in very large problems the computational cost of DMD itself may dominate, motivating compressed, projection-assisted, or incremental formulations that preserve enough reduced information to perform sparse amplitude selection afterward (Erichson et al., 2015, Matsumoto et al., 2017).

Even with these caveats, the method has had broad influence because it formalizes an issue that standard DMD left largely heuristic: modal selection. By turning mode selection into a convex optimization problem over amplitudes, SP-DMD made explicit the tradeoff between reconstruction fidelity and model complexity and provided a reproducible alternative to manual truncation (Jovanović et al., 2013). In Koopman-oriented extensions, the same idea now appears as sparse selection of informative Koopman triplets in EDMD and KDMD, often with additional pruning based on linear-evolution consistency (Pan et al., 2020). This suggests that SP-DMD is best understood not merely as a single algorithm, but as the prototype for a broader class of sparse post-processing methods applied to modal decompositions.

From this perspective, Sparsity-Promoting Dynamic Mode Decomposition occupies a specific position in the DMD family. It is neither a replacement for DMD nor simply a frequency filter. It is a sparse global amplitude-selection layer built on top of DMD, designed to produce compact, interpretable, and task-relevant reconstructions from otherwise unwieldy modal expansions (Jovanović et al., 2013).