Deep Learning-Based Signal Dimension Estimator

• DLSDE is a neural method that estimates the dimension of signals by interpreting intrinsic dimensions, model order, and spectral occupancy from raw observations.
• It integrates various deep learning architectures, including CNNs, autoencoders, and segmentation networks, to map raw or structured inputs to dimension-relevant quantities.
• The approach improves upon traditional eigenspectrum and feature-based estimators, enabling enhanced performance in radar detection, harmonic retrieval, and subspace learning.

A Deep Learning-Based Signal Dimension Estimator (DLSDE) is a neural estimator whose output is a task-specific notion of signal dimension. In the recent literature, that notion includes intrinsic dimensionality or number of latent variables in a set of measurements of a random vector, the number of paths or sources in a parametric signal model, and the spectral occupancy of a time-localized waveform as expressed by occupied FFT bins or effective bandwidth (Bahadur et al., 2019, Schieler et al., 2022, Ma et al., 14 Aug 2025, Pemasiri et al., 2024). DLSDE methods therefore appear in several adjacent areas—dimension estimation, model order selection, radar detection and localization, harmonic retrieval, and co-array subspace learning—while sharing a common objective: to replace or augment classical eigenspectrum- or feature-based estimators with learned mappings from raw observations or structured signal representations to dimension-relevant quantities.

1. Scope of the term and competing definitions

In the autoencoder literature, dimension estimation is defined as “the process of determining the intrinsic dimensionality or number of latent variables in a set of measurements of a random vector,” and is explicitly distinguished from dimension reduction, whose aim is to project data to a lower-dimensional space while preserving information (Bahadur et al., 2019). In this formulation, a DLSDE estimates the dimension of a lower-dimensional, possibly non-linear manifold embedded in a high-dimensional ambient space.

In phased-array radar, the same phrase is used differently. There, signal dimension is defined as the number of copies of a signal that arrive via different delays or angular shifts, so that correctly estimating the unknown $K$ becomes a prerequisite for high-resolution delay estimation and direction-finding algorithms such as MUSIC and sparse-reconstruction methods (Ma et al., 14 Aug 2025). In harmonic retrieval, an analogous quantity is the unknown number of deterministic, specular radio paths $P$, estimated jointly with continuous delay and Doppler parameters (Schieler et al., 2022).

A third usage appears in radar electronic-warfare processing. There, a segmentation network detects a radar signal-of-interest in raw complex I/Q samples, localizes it in time, and estimates the operating frequency band of the localized segment by applying an FFT to the masked waveform. In that setting, the spectral “dimension” can be defined as the number of occupied FFT bins or the bandwidth of the occupied band (Pemasiri et al., 2024).

A common misconception is that DLSDE always denotes source-count estimation. The literature instead uses the term for several related but non-identical inference problems: intrinsic dimension, model order, source number, and spectral occupancy. The unifying feature is not a single mathematical definition of “dimension,” but the use of a deep model to infer dimension-relevant structure directly from data.

2. Formal problem statements and signal models

Single-snapshot phased-array DLSDE begins from the observation model

$$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$

where $K$ is unknown, $a_k$ is the complex amplitude, $x_k$ is the key parameter per path, and $\boldsymbol{\varepsilon}$ is complex AWGN (Ma et al., 14 Aug 2025). For a ULA with inter-element spacing $d=\lambda/2$, the steering function is

$$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$

Traditional AIC-, MDL-, and GIC-based estimators operate on an empirical covariance constructed through spatial smoothing of a Hankel matrix, because the sample covariance from one snapshot is rank-1. The DLSDE in this setting instead uses the outer product $\mathbf{R}=\mathbf{r}\mathbf{r}^H$ directly as a 2D input and avoids EVD and smoothing (Ma et al., 14 Aug 2025).

In joint delay–Doppler harmonic retrieval, the discrete noiseless channel is

$P$0

with noisy observation $P$1, where $P$2 is unknown (Schieler et al., 2022). The DLSDE returns $P$3, $P$4, and $P$5, while $P$6 is obtained by least-squares/BLUE after the kinematic parameters are estimated.

In sparse linear arrays, the data model is covariance-centric. A virtual ULA covariance $P$7 is associated with signal and noise subspaces $P$8 and $P$9, with orthogonal projectors

$$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$0

Here, DLSDE is closely tied to inferring the number of sources $$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$1 through learned subspace representations on Grassmannians, particularly in co-array settings that localize more sources than physical sensors (Chen et al., 2024).

In radar signal detection and FFT estimation, the primary object is a length-$$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$2 complex I/Q sequence. A per-sample mask $$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$3 identifies signal presence, and the masked signal

$$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$4

is transformed by the DFT

$$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$5

If the predicted FFT magnitude $$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$6 is thresholded to obtain occupied bins $$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$7, then

$$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$8

so the model yields a spectral-dimension estimate together with temporal localization (Pemasiri et al., 2024).

These formulations show that DLSDE is not restricted to one observation regime. It can operate on raw I/Q sequences, outer products of single snapshots, sample covariances, or latent matrices extracted by autoencoders.

3. Learned representations and model architectures

One architectural class uses segmentation networks over time-domain signals. For radar I/Q input, the core model is a 1D U-Net with a contracting path, an expansive path, skip connections, and Double Convolution blocks of the form $$\mathbf{r} = \sum_{k=1}^{K} a_k \, \mathbf{b}(x_k) + \boldsymbol{\varepsilon},$$9. The input is a two-channel $K$0 sequence of fixed length $K$1, and the main output is a per-sample segmentation probability $K$2 (Pemasiri et al., 2024). There is not an explicit dual-branch fusion network; frequency-domain information is coupled implicitly through the loss.

A second class uses autoencoders for intrinsic-dimension estimation. Bahadur and Paffenroth study a vanilla autoencoder with latent-layer sparsity, $K$3 normalization of latent activations, and the explicit recommendation to avoid an activation function immediately before the innermost hidden layer so that scale information is preserved for dimension estimation (Bahadur et al., 2019). Their dimension estimate is derived from singular value proxies constructed from the hidden-layer matrix by taking absolute values, sorting each row, and averaging columns.

A related but structurally different design is the additive autoencoder. It serially performs bias estimation and scaling, linear trend estimation via PCA, and nonlinear residual estimation with a feedforward autoencoder using tanh in hidden layers and a final linear activation. The additive reconstruction is

$K$4

The paper compares shallow and deeper symmetric networks, including 1Sym, 3Sym, 5Sym, and 7Sym, where deeper symmetric architectures enlarge hidden widths systematically while preserving a bottleneck interpretation (Kärkkäinen et al., 2022).

Model-order DLSDEs in sensing and communications often use 2D CNNs. In the phased-array single-snapshot setting, the input is a $K$5 tensor formed from $K$6 and $K$7. The network is a 5-layer 2D-CNN with three convolutional blocks, max pooling after each convolution, dropout after the third block, a dense layer of size 1024, and a final dense layer mapping to $K$8 classes, with $K$9 in the experiments (Ma et al., 14 Aug 2025).

The harmonic-retrieval architecture is more elaborate. It constructs $a_k$0 windowed views—Tukey, Taylor, Chebyshev, Blackman, Flat Top, Cosine, Hann, and Rectangular—applies a 2D-DFT, and converts each complex view into four real feature channels,

$a_k$1

yielding a $a_k$2-channel CNN input (Schieler et al., 2022). The network then predicts cell-wise objectness flags and continuous offsets in a quasi-grid-free parameterization, together with a dedicated model-order head.

Subspace-learning DLSDEs use representation learning on Grassmannians. The sparse-array methodology employs a WRN-16-8 without batch normalization, uses the real and imaginary parts of the complex sample covariance as input channels, outputs a complex matrix $a_k$3, and forms $a_k$4 so that its leading eigenvectors define a basis-invariant learned signal subspace (Chen et al., 2024). The representation is intentionally invariant to the selection of bases.

4. Objectives, estimators, and decision rules

In radar segmentation and spectral estimation, the training objective is the compound loss

$a_k$5

The segmentation term is binary cross-entropy, and the FFT term is mean squared error between normalized ground-truth and predicted FFT magnitudes. The paper states that $a_k$6 penalizes spectral discrepancies and encourages the segmentation network to be temporally precise where the signal energy resides (Pemasiri et al., 2024).

In the autoencoder formulation of Bahadur and Paffenroth, the basic objective is reconstruction,

$a_k$7

augmented by a sparsity penalty on normalized latent activations (Bahadur et al., 2019). Dimension is not read directly from a classifier. Instead, the latent matrix is transformed into singular value proxies, and two estimators are applied: a “Greater Than Equal To 1%” rule that counts how many singular values contribute at least $a_k$8 of the total sum, and an “Up to 90%” rule that chooses the smallest number of largest singular values whose squared contributions reach at least $a_k$9 of the total squared sum.

The additive-autoencoder approach uses a reconstruction-error sweep over bottleneck sizes. Its criterion is the mean root squared error

$x_k$0

and the intrinsic dimension is selected from the first bottleneck size $x_k$1 such that the backward difference $x_k$2 falls below a threshold $x_k$3, with the reported rule $x_k$4 (Kärkkäinen et al., 2022).

In quasi-grid-free harmonic retrieval, the total loss is

$x_k$5

where $x_k$6 is BCE on the one-hot model-order output and $x_k$7 is a masked MSE on delay and Doppler offsets, gated by a sigmoid-transformed objectness variable (Schieler et al., 2022). At inference, $x_k$8 is obtained by argmax over the order head, while per-cell components are retained when the objectness exceeds a threshold.

The phased-array single-snapshot DLSDE uses a standard cross-entropy classification objective over the classes $x_k$9, and its inference pipeline is deliberately simple: form $\boldsymbol{\varepsilon}$0, split real and imaginary parts into two channels, pass the tensor through the CNN, apply softmax, and take $\boldsymbol{\varepsilon}$1 (Ma et al., 14 Aug 2025). No smoothing or post-processing thresholding is used beyond the classification.

In subspace representation learning, the losses are geometric. The geodesic distance on $\boldsymbol{\varepsilon}$2 is

$\boldsymbol{\varepsilon}$3

and the chordal distance is

$\boldsymbol{\varepsilon}$4

with principal angles defined by $\boldsymbol{\varepsilon}$5 (Chen et al., 2024). For variable-dimension learning, the paper extends the comparison to a union of Grassmannians and accelerates training by consistent rank sampling, in which all examples in a minibatch share the same subspace dimension.

5. Empirical performance across domains

Because the target variable changes across papers, reported metrics are not directly comparable. The literature evaluates DLSDE by F1 and cosine similarity in radar segmentation, successful detection rate and angular resolution in single-snapshot phased arrays, model-order error and runtime in harmonic retrieval, and latent-dimension estimates or reconstruction-error knees in autoencoder studies.

Setting	Representative findings	Reference
EW radar detection and FFT estimation	At $\boldsymbol{\varepsilon}$6 dB, $\boldsymbol{\varepsilon}$7 improves from $\boldsymbol{\varepsilon}$8 to $\boldsymbol{\varepsilon}$9–$d=\lambda/2$0; at $d=\lambda/2$1 dB, FFT cosine similarity rises from $d=\lambda/2$2 to $d=\lambda/2$3–$d=\lambda/2$4; the energy detector achieves $d=\lambda/2$5, $d=\lambda/2$6, versus $d=\lambda/2$7, $d=\lambda/2$8 for the proposed model	(Pemasiri et al., 2024)
Joint delay–Doppler retrieval and model order	The CNN’s $d=\lambda/2$9 is consistently best across SNRs, particularly at low SNR; runtime is $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$0 ms/sample for the CNN, $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$1 ms/sample for CNN + 10 Gauss–Newton iterations, and $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$2 s/sample for RIMAX	(Schieler et al., 2022)
Single-snapshot phased-array radar	DLSDE reaches $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$3 successful detection at SNR $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$4 dB, while conventional estimators achieve $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$5 even at $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$6 dB; it achieves $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$7 successful detection at $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$8, whereas conventional estimators require $$y_i(x)=\exp\!\left(j \, \frac{2\pi}{\lambda} \, (i-1)\, d \, \sin x\right), \quad i=1,\ldots,N.$$9 and GIC reaches $\mathbf{R}=\mathbf{r}\mathbf{r}^H$0 at $\mathbf{R}=\mathbf{r}\mathbf{r}^H$1	(Ma et al., 14 Aug 2025)
Autoencoder intrinsic-dimension estimation	On MNIST, AE “Up to 90%” estimates are around $\mathbf{R}=\mathbf{r}\mathbf{r}^H$2–$\mathbf{R}=\mathbf{r}\mathbf{r}^H$3 and AE “$\mathbf{R}=\mathbf{r}\mathbf{r}^H$4” estimates are roughly $\mathbf{R}=\mathbf{r}\mathbf{r}^H$5–$\mathbf{R}=\mathbf{r}\mathbf{r}^H$6; in the S&P 500 time series, estimated dimension drops during market stress	(Bahadur et al., 2019)
Additive autoencoder dimension estimation	Mean efficiencies relative to 1Hid are $\mathbf{R}=\mathbf{r}\mathbf{r}^H$7 for 1Sym, $\mathbf{R}=\mathbf{r}\mathbf{r}^H$8 for 3Sym, $\mathbf{R}=\mathbf{r}\mathbf{r}^H$9 for 5Sym, and $P$00 for 7Sym; deeper networks obtain lower autoencoding errors during the search, but the detected dimension does not change compared to a shallow network	(Kärkkäinen et al., 2022)
Sparse-array subspace learning	Learned subspace representations outperform SPA and existing DNN-based covariance reconstruction methods for a wide range of SNRs, snapshots, and source numbers for both perfect and imperfect arrays	(Chen et al., 2024)

Two patterns recur. First, deep estimators are most competitive when the classical baseline depends on fragile intermediate quantities such as a poorly estimated covariance, a discretized grid, or hand-designed statistics. Second, many successful DLSDEs are hybrid rather than purely end-to-end: the harmonic-retrieval network supplies a warm start for Gauss–Newton refinement, and the radar segmentation model uses a neural mask but a deterministic FFT to derive the band estimate.

6. Limitations, interpretation, and active directions

The main interpretive limitation is semantic rather than algorithmic: “dimension” is not uniform across the literature. It may denote intrinsic manifold dimension, model order, source count, or spectral occupancy. Any comparison of DLSDE methods therefore depends on whether the underlying target is latent, parametric, or spectral.

Task-specific limitations are substantial. In radar FFT estimation, performance at very low SNR is degraded by missed detections, imprecise boundaries, and leakage from the rectangular window implied by the segmentation mask; band-edge accuracy depends on threshold selection, and adaptive FFT sizing or multi-resolution STFT was not explored (Pemasiri et al., 2024). In autoencoder-based DE, the methods are data hungry, sensitive to the sparsity parameter $P$01, and the 2019 study provides no formal proofs of consistency or error bounds (Bahadur et al., 2019). In the additive-autoencoder study, deeper symmetric models lower MRSE during the search but do not change the detected intrinsic dimension, and threshold choice for $P$02MRSE remains data-scale dependent (Kärkkäinen et al., 2022).

Model-order DLSDEs also inherit modeling assumptions. The harmonic-retrieval method assumes a narrowband model, a single snapshot, and synthetic data without hardware impairments; finite per-cell capacity can limit dense local multiplicities, and the raw CNN saturates before high-resolution ML accuracy unless refinement is applied (Schieler et al., 2022). The phased-array single-snapshot classifier was trained at a fixed high SNR of $P$03 dB because mixing low-SNR examples degraded training stability, and the reported experiments restrict $P$04 to $P$05 under a ULA, AWGN, and CDL channel model (Ma et al., 14 Aug 2025). In sparse-array subspace learning, extremely low SNR, very few snapshots, and highly coherent sources can blur projector spectra and complicate rank decisions (Chen et al., 2024).

A second misconception is that deeper architectures necessarily yield better dimension estimates. The additive-autoencoder results explicitly separate reconstruction quality from the detected dimension: deeper models obtain lower autoencoding errors during the identification of the intrinsic dimension, but the detected dimension does not change compared to a shallow network (Kärkkäinen et al., 2022). Likewise, the harmonic-retrieval literature indicates that high-precision parameter estimation may still require classical ML refinement even when model order is estimated well by the network (Schieler et al., 2022).

This suggests that DLSDE is best understood as a learned front end for extracting dimension-relevant structure, not as a universal replacement for all downstream inference. A plausible implication is that the most durable designs will continue to combine learned representations with explicit signal models: masked FFTs for spectral occupancy, least-squares or BLUE for gain recovery, Gauss–Newton refinement for quasi-grid-free estimates, and projector-based geometry for source-number inference.