Sparse Separable Factor Analysis in the Complex Domain with an Application to Local Field Potential Data

Ian Hultman; Kirtikanth Kalapatapu; Yassine Filali; Rainbo Hultman; Sanvesh Srivastava

stat.MLAug 21, 2026

Sparse Separable Factor Analysis in the Complex Domain with an Application to Local Field Potential Data

Authors: Ian Hultman, Kirtikanth Kalapatapu, Yassine Filali, Rainbo Hultman, Sanvesh Srivastava

Organizations: Department of Statistics and Actuarial Science, University of Iowa · Department of Molecular Physiology and Biophysics, University of Iowa

Abstract

Complex-valued arrays arise in signal processing, where scientific interpretation depends on retaining amplitude and phase information. Existing covariance estimation methods either ignore the multiway organization of such data or rely on real-domain embeddings that do not directly exploit their complex structure. We develop sparse separable factor analysis (SSFA), a latent factor model for complex-valued arrays with a separable covariance structure across modes. Each mode-specific covariance matrix is modeled through a low-rank Hermitian factor structure and a diagonal residual covariance matrix. To obtain interpretable estimates, we impose elementwise lasso penalties on the complex loading matrices and estimate the SSFA parameters using a mode-wise parameter-expanded expectation-maximization procedure. The resulting loading updates admit closed-form complex soft-thresholding solutions, which shrink the modulus of each loading while preserving its phase. A separate balancing step resolves the scale nonidentifiability of the separable covariance structure. Simulation studies show that SSFA improves covariance estimation relative to vectorization-based methods, including complex principal component analysis. We apply SSFA to local field potential recordings from mice, where we compare separability structures induced by different groupings of brain region, frequency, and time and perform model-based imputation of recordings missing because of electrode misplacement.

Figures & tables

Figure 1: Estimated coherence matrices (top panels) and phase-offset matrices (bottom panels) at 1, 2, 5, and 20 Hz from the SSFA model fitted to the full 600-second time series in Section 6.3 .

Figure 2: Estimated coherence matrices at 1 and 2 Hz (first two panels) and the estimated canonical cross-coherence matrix between 1 and 2 Hz (third panel) from the SSFA model fitted to the full 600-second time series in Section 6.3 .

Setting	Observation type	Observation size	Dimension vector $\mathbf{p}$	Rank vector $\mathbf{k}$
1	Vector	$\xb_{i}\in\CC^{25}$	$(25)$	$(3)$
2	Vector	$\xb_{i}\in\CC^{50}$	$(50)$	$(4)$
3	Matrix	$\Xb_{i}\in\CC^{25\times 25}$	$(25,25)$	$(4,3)$
4	Matrix	$\Xb_{i}\in\CC^{50\times 50}$	$(50,50)$	$(4,3)$
5	Array	$\Xcal_{i}\in\CC^{25\times 25\times 25}$	$(25,25,25)$	$(4,3,2)$
6	Array	$\Xcal_{i}\in\CC^{50\times 50\times 50}$	$(50,50,50)$	$(4,3,2)$

Table 1: Simulation settings. For each setting,

\Lambdab_{j}\in\CC^{p_{j}\times k_{j}}

and

\Psib_{j}\in\RR_{+}^{p_{j}\times p_{j}}

is diagonal for

j=1,\ldots,d

Panel of Figure 3 — Figure 3: Example of the moduli of a rank-three sparse loading matrix and the corresponding mode-specific covariance matrix used in the simulation study.

Setting	Size	Complex PCA	Sparse PCA	SSFA	Empirical
$\xb_{i}\in\CC^{25}$	$n=5$	0.828 (0.154)	1.058 (0.219)	0.811 (0.148)	0.813 (0.144)
	$n=10$	0.597 (0.103)	0.651 (0.149)	0.590 (0.097)	0.598 (0.100)
	$n=20$	0.427 (0.070)	0.380 (0.065)	0.419 (0.063)	0.429 (0.063)
	$n=40$	0.303 (0.055)	0.276 (0.039)	0.301 (0.052)	0.307 (0.051)
	$n=100$	0.197 (0.027)	0.179 (0.030)	0.191 (0.028)	0.199 (0.026)
$\xb_{i}\in\CC^{50}$	$n=5$	0.994 (0.186)	1.620 (0.316)	0.939 (0.177)	0.942 (0.181)

Table 2: Comparison across methods of the median and median absolute deviation of

\text{err}_{\Sigmab}

defined in ( 39 ). The median absolute deviations are in parentheses. Empirical denotes the empirical covariance estimation method. A “–” indicates that the entry could not be computed because of prohibitive computational cost.

$\mathrm{err}_{\Pb_{1}}$
Setting	Size	Complex PCA	Sparse PCA	SSFA
$\xb_{i}\in\CC^{25}$	$n=5$	0.664 (0.089)	1.358 (0.108)	0.664 (0.086)
	$n=10$	0.389 (0.049)	1.021 (0.405)	0.388 (0.038)
	$n=20$	0.275 (0.029)	0.299 (0.031)	0.258 (0.029)
	$n=40$	0.191 (0.017)	0.211 (0.022)	0.174 (0.016)
	$n=100$	0.117 (0.010)	0.136 (0.012)	0.103 (0.009)

Table 3: Median and median absolute deviation of the loading-matrix errors

\mathrm{err}_{\Pb_{1}}

and

\mathrm{err}_{\Lambdab_{1}}

defined in ( 37 ) and ( 38 ) for the vector-valued simulations. Median absolute deviations are shown in parentheses.

scenario	size	$\text{err}_{\Sigmab_{1}}$	$\text{err}_{\Lambdab_{1}}$	$\text{err}_{\Pb_{1}}$	proportion zero $\widehat{\Lambdab}_{1}$	rank $\widehat{\Lambdab}_{1}$	true rank $\Lambdab_{1}$
$\xb_{i}\in\CC^{25}$	$n=5$	0.811 (0.148)	0.383 (0.050)	0.664 (0.086)	0.000 (0.000)	4.000 (0.000)	3
	$n=10$	0.590 (0.097)	0.218 (0.024)	0.388 (0.038)	0.040 (0.016)	5.000 (0.000)
	$n=20$	0.419 (0.063)	0.142 (0.017)	0.258 (0.029)	0.056 (0.008)	5.000 (0.000)
	$n=40$	0.301 (0.052)	0.098 (0.009)	0.174 (0.016)	0.064 (0.016)	5.000 (0.000)
	$n=100$	0.191 (0.028)	0.057 (0.004)	0.103 (0.009)	0.096 (0.024)	5.000 (0.000)
$\xb_{i}\in\CC^{50}$	$n=5$	0.939 (0.177)	0.486 (0.047)	0.984 (0.101)	0.000 (0.000)	4.000 (0.000)	4

Table 4: SSFA mode-one estimation results. Entries report medians, with median absolute deviations shown in parentheses. The metrics

\mathrm{err}_{\Sigmab_{1}}

\mathrm{err}_{\Lambdab_{1}}

, and

\mathrm{err}_{\Pb_{1}}

are defined in ( 39 ), ( 38 ), and ( 37 ), respectively.

scenario	size	$\text{err}_{\Sigmab_{2}}$	$\text{err}_{\Lambdab_{2}}$	$\text{err}_{\Pb_{2}}$	proportion zero $\widehat{\Lambdab}_{2}$	rank $\widehat{\Lambdab}_{2}$	true rank $\Lambdab_{2}$
$\Xb_{i}\in\CC^{25\times 25}$	$n=5$	0.206 (0.035)	0.054 (0.005)	0.097 (0.008)	0.088 (0.024)	5.000 (0.000)	3
	$n=10$	0.142 (0.032)	0.037 (0.004)	0.065 (0.007)	0.111 (0.031)	4.000 (1.000)
	$n=20$	0.104 (0.025)	0.025 (0.002)	0.045 (0.004)	0.120 (0.027)	3.000 (0.000)
	$n=40$	0.078 (0.016)	0.017 (0.001)	0.031 (0.003)	0.120 (0.027)	3.000 (0.000)
$\Xb_{i}\in\CC^{50\times 50}$	$n=5$	0.147 (0.028)	0.039 (0.002)	0.068 (0.003)	0.096 (0.044)	5.000 (0.000)	3
	$n=10$	0.112 (0.025)	0.026 (0.001)	0.045 (0.003)	0.153 (0.037)	3.000 (0.000)

Table 5: SSFA mode-two estimation results. Entries report medians, with median absolute deviations shown in parentheses. The metrics

\mathrm{err}_{\Sigmab_{2}}

\mathrm{err}_{\Lambdab_{2}}

, and

\mathrm{err}_{\Pb_{2}}

are defined in ( 39 ), ( 38 ), and ( 37 ), respectively.

scenario	size	$\text{err}_{\Sigmab_{3}}$	$\text{err}_{\Lambdab_{3}}$	$\text{err}_{\Pb_{3}}$	proportion zero $\widehat{\Lambdab}_{3}$	rank $\widehat{\Lambdab}_{3}$	true rank $\Lambdab_{3}$
$\Xcal_{i}\in\CC^{25\times 25\times 25}$	$n=5$	0.052 (0.021)	0.009 (0.001)	0.013 (0.001)	0.160 (0.040)	2.000 (0.000)	2
	$n=10$	0.048 (0.020)	0.007 (0.001)	0.009 (0.001)	0.200 (0.060)	2.000 (0.000)
	$n=20$	0.047 (0.021)	0.005 (0.000)	0.007 (0.001)	0.280 (0.060)	2.000 (0.000)
	$n=40$	0.046 (0.022)	0.004 (0.001)	0.005 (0.001)	0.340 (0.080)	2.000 (0.000)
$\Xcal_{i}\in\CC^{50\times 50\times 50}$	$n=5$	0.050 (0.018)	0.005 (0.000)	0.007 (0.001)	0.280 (0.065)	2.000 (0.000)	2
	$n=10$	0.042 (0.019)	0.004 (0.001)	0.005 (0.001)	0.355 (0.065)	2.000 (0.000)

Table 6: SSFA mode-three estimation results. Entries report medians, with median absolute deviations shown in parentheses. The metrics

\mathrm{err}_{\Sigmab_{3}}

\mathrm{err}_{\Lambdab_{3}}

, and

\mathrm{err}_{\Pb_{3}}

are defined in ( 39 ), ( 38 ), and ( 37 ), respectively.

Table 7: Median log relative norm error

\mathrm{logerr}_{m,r}

from ( 41 ), stratified by brain region. Smaller values indicate better imputation. Results are shown for mean imputation, empirical-covariance imputation, and the unpenalized vectorized complex factor model with

k\in\{10,25,50,100,150,250\}

. Median absolute deviations are reported in parentheses.

Figure 4: Brain-region mode-specific coherence and phase-offset matrices from the fully separable SSFA model fitted to $\Xcal_{\mathrm{arr}}$ . The model was fitted to the post-administration data under both the treatment and control conditions.

Figure 5: Moduli of the two largest-magnitude time-mode loading vectors from the SSFA fits to

\Xcal_{\mathrm{mat}}

and

\Xcal_{\mathrm{arr}}

. The models were fitted to the post-administration data under both the treatment and control conditions. The first and final 300 seconds correspond to phases 1 and 2, respectively.

Figure 6: Estimated coherence matrices at selected frequencies from SSFA, complex PCA, and sparse PCA fitted to $\Xb_{\mathrm{vec}}$ . The comparison uses phase 1 of the post-administration period from the two control days.

Figure 7: Estimated phase-offset matrices at selected frequencies from SSFA, complex PCA, and sparse PCA fitted to $\Xb_{\mathrm{vec}}$ . The comparison uses phase 1 of the post-administration period from the two control days.

Figure 8: Elementwise moduli of the normalized cross-frequency correlation blocks between the brain-region features at 1 Hz and 2 Hz. Estimates are shown for SSFA, complex PCA, and sparse PCA fitted to $\Xb_{\mathrm{vec}}$ .

Figure 9: Moduli of the six largest-magnitude brain-region–frequency loading vectors from SSFA, complex PCA, and sparse PCA fitted to $\Xb_{\mathrm{vec}}$ . Loading entries are shown for the first seven frequencies within each brain region.

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May 26, 2026J/K move · Enter open · S save

When do complex-valued neural networks help? A study of representation, geometry, and optimization

Ashutosh Kumar

Owl Autonomous Imaging, Inc.

Neural Network Optimization Complex-Valued Neural Networks Neural Network Training Dynamics

Open paper AskPDF arXiv

Complex-valued Neural Networks (CVNNs) are often motivated by domains where information is naturally encoded in magnitude and phase. Yet complex-valued inputs alone do not determine when complex arithmetic improves learning: the label signal may lie in amplitude, phase, their coupling, or a symmetry that real-valued models can also represent under suitable coordinates. We study this through a representation-first evaluation of CVNNs against Cartesian real, polar, phase-only, magnitude-only, parameter-matched real, and FLOP-matched real baselines. Across synthetic RF tasks, complex representations are useful but not universally superior. PSK-only tasks favor phase-aware and complex-valued models, QAM-only tasks favor magnitude-based models, mixed PSK+QAM gives only a small complex-valued advantage, and unseen carrier-phase rotations break coordinate-dependent models without augmentation. Similar patterns appear beyond RF: in quantum-wavefunction prediction, momentum is invisible to

|ψ|

but recoverable from phase, while EEG analytic-signal experiments show that phase locking, amplitude bursts, and phase-amplitude coupling each favor different coordinate views. We also identify a benchmarking artifact on RadioML 2018.01A. Under matched-shared-trial selection, a CReLU complex model exceeds the best real baseline by 22.94 PP; under independent per-family tuning on the same data and 16-trial search space, the gap collapses to 2.46 PP. Gradient analysis traces the inflated gap to high-learning-rate first-step instability in real baselines, while complex parameter coupling distributes the loss signal more robustly. A learning-rate

\times

activation factorial confirms the failure is primarily hyperparameter-driven. Overall, CVNNs are best viewed as structured inductive biases whose gains depend on representation, symmetry, and optimization, not as universally superior architectures.

Abstract

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