Disorder-Enabled Joint Projection

• Disorder-enabled joint projection is a framework that uses controlled randomness to embed multiple functions or representations into a shared physical or latent space.
• It is applied across photonic lattices for dimensionality reduction, synthetic metasurfaces for optical multiplexing, and neuroimaging for joint feature alignment.
• Methodologies leverage statistical guarantees, engineered disorder, and contrastive learning to preserve task-relevant structure, improving imaging, signal processing, and classification outcomes.

Searching arXiv for the cited works and closely related terminology to ground the article in current papers. arXiv search: "Disorder-enabled Synthetic Metasurfaces" Disorder-enabled joint projection denotes a class of mappings in which disorder, heterogeneous allocation, or cross-view alignment is used to project multiple functions, signals, or representations into a shared physical or latent substrate. In the recent literature considered here, the term appears in three distinct settings: disordered photonic lattices that realize random projections satisfying Johnson–Lindenstrauss-type guarantees (Miri, 2021), synthetic metasurfaces that use engineered structural disorder to encode multiple optical functions in a single aperture and jointly project them in real and momentum space (Li et al., 7 Jul 2025), and cross-view contrastive learning that aligns volumetric imaging and ROI-graph embeddings for brain disorder classification (Liang et al., 10 Mar 2026). The common technical motif is not a single formalism but a shared objective: preserve task-relevant structure while jointly embedding multiple degrees of freedom into a compact projector.

1. Scope and terminological usage

In the cited literature, “disorder-enabled joint projection” is associated with three different projection targets: lower-dimensional vector embeddings, multifunctional optical fields, and shared latent representations for classification. This suggests a family resemblance rather than a single standardized doctrine.

Domain	Projected entities	Projection mechanism
Disordered photonic lattices	Vectors in $\mathbb{C}^N$	Row-sampled unitary evolution with intermediate diagonal disorder
Synthetic metasurfaces	Multiple optical functions	Disordered meta-pixel shares with qBIC-based selectivity
Brain disorder classification	Imaging and ROI embeddings	Bidirectional cross-view contrastive alignment

A notable terminological distinction is that, in the photonic works, “disorder” refers to engineered physical randomness or structural nonuniformity, whereas in the neuroimaging work the term is tied to brain disorder classification rather than to physical disorder (Liang et al., 10 Mar 2026). The shared word “projection” is likewise domain-specific: in photonic lattices it denotes a dimensionality-reducing linear map, in metasurfaces a far-field reconstruction from selected meta-pixel subsets, and in neuroimaging a learned map into a common latent space.

2. Random complex projections in disordered photonic lattices

In the photonic-lattice formulation, one considers an array of $N$ single-mode waveguides with nearest-neighbor evanescent coupling and diagonal disorder. The coupled-mode dynamics are

$$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$

with tight-binding Hamiltonian

$$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$

Here $\kappa$ is the uniform coupling constant, and each detuning $\delta_i$ is drawn i.i.d. from a zero-mean distribution of width $\sigma$. Over propagation length $L$, the evolution operator is the unitary matrix

$U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$

In the eigenmode basis, the matrix elements satisfy

$$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$

When the disorder is neither too weak nor too strong, the sum over modes becomes well mixed, and the entries of $N$0 approach circular complex Gaussians:

$N$1

This Gaussianization is the key step that connects physical propagation to random projection theory (Miri, 2021).

The resulting dimensionality reduction is implemented by row-sampling the unitary. If $N$2 is the $N$3 matrix formed by selecting $N$4 output channels out of $N$5, then $N$6 acts as a partial-unitary embedding. The cited formulation gives a complex Johnson–Lindenstrauss statement: for a set of $N$7 vectors in $N$8, if $N$9 has i.i.d. $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$0 entries and

$$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$1

then with probability at least $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$2 all pairwise distances are preserved within a factor $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$3. Under the approximation $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$4 and near-orthonormal sampled rows, the same argument applies to $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$5. For any fixed $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$6,

$$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$7

so the embedding dimension scales as $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$8 (Miri, 2021).

3. Disorder regime, transport physics, and implementation constraints

The same work distinguishes three transport regimes as $$\frac{d}{dz}\mathbf{u}(z) \;=\; -\,i\,H\,\mathbf{u}(z), \qquad \mathbf{u}=(u_1,\dots,u_N)^T,$$9 increases. In the ballistic regime, $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$0, disorder is too weak: eigenstates are extended plane waves, phases remain highly structured, and the matrix entries are neither zero-mean nor Gaussian. In the localized regime, $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$1, Anderson localization dominates, each input excites only a few nearby sites, and the columns of $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$2 become sparse. The desired projection behavior therefore requires an intermediate diffusive regime, $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$3, in which wave interference over many scattering events yields random-matrix-type mixing (Miri, 2021).

The stated conditions for Johnson–Lindenstrauss behavior are

$$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$4

together with the practical inequality

$$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$5

The article further reports that one may sweep $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$6 and verify Gaussianity of $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$7 באמצעות a Kolmogorov–Smirnov test. Since $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$8, the output is rescaled by $$H_{ij} \;=\; \begin{cases} \delta_i, & i=j,\ \kappa, & |i-j|=1\text{ (nearest neighbors)},\ 0, & \text{otherwise.} \end{cases}$$9 to compensate average power loss (Miri, 2021).

Several operational modes are described. In time-multiplexing, a train of $\kappa$0 short pulses carries the inputs $\kappa$1, and the $\kappa$2th time bin at the output contains $\kappa$3. In multi-channel excitation, disjoint input waveguides or orthogonal modulation codes suppress cross-talk when signals are sparse in time or wavelength. Correlated inputs remain compatible with the guarantee because the Johnson–Lindenstrauss bound is uniform over all pairs; clustered inputs remain clustered after projection. A common misconception, explicitly contradicted by the regime analysis, is that increasing disorder monotonically improves randomness. The cited result instead requires intermediate diagonal disorder: too little disorder fails to randomize, while too much induces localization (Miri, 2021).

4. Engineered disorder in synthetic metasurfaces

The metasurface formulation treats disorder as a design variable that enables many optical functions to coexist within a single aperture. The proposed optimization balances per-function fidelity with disorder regularization:

$\kappa$4

where $\kappa$5 is the simulated field or phase distribution for function $\kappa$6, $\kappa$7 is the target profile, $\kappa$8 are the meta-pixel coordinates, and $\kappa$9 penalizes too much clustering or too much ordering. The regularizer is given as

$\delta_i$0

Varying $\delta_i$1, $\delta_i$2, and $\delta_i$3 tunes the pattern between minimum-clustering and overly sparse arrangements, while $\delta_i$4 balances function fidelity against disorder quality (Li et al., 7 Jul 2025).

Spectral selectivity is implemented through nonlocal meta-pixels engineered to support quasi-bound states in the continuum. In an idealized lossless structure, a BIC has $\delta_i$5; slight symmetry breaking yields a qBIC resonance with

$\delta_i$6

The transmission coefficient is modeled as

$\delta_i$7

with $\delta_i$8. By rotating each T-shaped meta-pixel in-plane, the geometric phase $\delta_i$9 can be set independently without degrading the $\sigma$0-factor (Li et al., 7 Jul 2025).

A central empirical claim is that a single optical function does not require a contiguous aperture. Random sub-sampling to a fraction $\sigma$1 yields a functional density $\sigma$2. Using the angular-spectrum method numerically and the Strehl ratio experimentally, the study reports that disordered sampling maintains $\sigma$3 down to $\sigma$4, whereas an ordered sector-shaped aperture with the same $\sigma$5 already fails, reaching $\sigma$6 by $\sigma$7. Disorder is quantified using Moran’s index,

$\sigma$8

and the Strehl ratio is reported to recover suddenly as $\sigma$9 drops from $L$0 to $L$1. The filling factor is also constrained by a spatial Nyquist condition,

$L$2

where $L$3 is the maximum spatial frequency of the wavefront. In the demonstrated platform, each meta-pixel is $L$4, and at $L$5 the design achieves $L$6 while avoiding Bragg diffraction orders (Li et al., 7 Jul 2025).

5. Real-space and momentum-space joint projection

The metasurface proof of concept combines spectral multiplexing of lens profiles with polarization multiplexing of gratings. For the spectral channel, $L$7 wavelengths $L$8 are each assigned a randomly placed subset of meta-pixels. A pixel at position $L$9 carries a resonance tuned to $U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$0 and a rotation encoding the lens phase

$U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$1

Because of the high $U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$2-factor, the transmission is approximated by

$U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$3

Subsampling to $U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$4 for each wavelength leaves the remaining area available for the other ten functions (Li et al., 7 Jul 2025).

For polarization multiplexing, three disordered shares, each with $U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$5, are assigned to momentum-space gratings corresponding to the orthogonal bases horizontal/vertical, diagonal/anti-diagonal, and right/left circular. On resonance for polarization basis $U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$6, the transmission is

$U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$7

with $U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$8 the grating wave vector; for example, H is deflected to $U \;=\;\exp\!\bigl(-i\,H\,L\bigr)\;\in U(N).$9 and V to $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$0. The full far-field intensity is

$$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$1

When wavelength and polarization select a single share, the other contributions vanish approximately and the target lens or grating is reconstructed in one shot (Li et al., 7 Jul 2025).

The reported demonstration includes a synthetic achromatic metalens with aperture diameter $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$2, focal length $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$3, and $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$4, corresponding to a diffraction-limited spot of approximately $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$5. The measured Strehl ratio is $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$6 over $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$7–$$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$8, and the chromatic focal shift is $$U_{ij} \;=\; \sum_{n=1}^N e^{-i\,\beta_n L}\;\phi^{(n)}_i\,\phi^{(n)*}_j.$$9 of $N$00, compared with a $N$01 shift for a single-wavelength reference lens. The experimentally measured resonance quality factor is $N$02, while simulations and deeper patterning can reach $N$03 (Li et al., 7 Jul 2025).

The same platform supports single-shot polarimetric imaging with minimum super-pixel size $N$04, spatial resolution of approximately $N$05, and polarization reconstruction error $N$06. The work reports imaging of radially and azimuthally polarized vector beams, whose raw $N$07-space spots appear as six lobes mapping exactly to H/V, D/A, and R/L content, as well as characterization of an optical skyrmion with topological number approximately $N$08 in a single acquisition (Li et al., 7 Jul 2025).

The same framework is explicitly extended in the cited discussion to arbitrary hologram multiplexing, OAM holography using helical phases $N$09, and higher-dimensional joint control over wavelength, polarization, OAM, angle of incidence, and temporal waveforms. Trade-offs are also stated: cross-talk increases when resonances overlap, higher $N$10 tightens spectral isolation but enlarges pixel footprints and slows response, disorder-induced speckle or side-lobes can be reduced with a band-limited regularizer, and fabrication tolerances may be compensated by in-line metrology and neural-network-based correction (Li et al., 7 Jul 2025).

6. Cross-view joint projection for brain disorder classification

In neuroimaging, Liang and He describe a joint imaging-ROI representation learner in which volumetric and graph-based subject representations are projected into a shared latent space through bidirectional contrastive alignment (Liang et al., 10 Mar 2026). The imaging branch takes a preprocessed $N$11D T1-weighted MRI volume $N$12 and applies a 3DSC-TF encoder, described as a hybrid 3D depthwise-separable CNN plus Transformer, to produce a $N$13-dimensional embedding. The ROI branch takes a subject-specific graph $N$14, where $N$15 are AAL parcel mean intensities and $N$16 are Pearson correlations, and uses a GNN called “NeuroGraph” with $N$17 message-passing layers and hidden size $N$18 to produce a $N$19-dimensional embedding. Two projection heads, both two-layer MLPs, map these embeddings to a shared dimension $N$20:

$N$21

$N$22

For a mini-batch of size $N$23, the similarity matrix is

$N$24

with cosine similarity $N$25 and temperature $N$26. The bidirectional InfoNCE-style objective is

$N$27

Positive pairs are same-subject imaging and ROI embeddings, and all cross-subject pairs are negatives. After alignment, the projection heads are discarded and the base embeddings are concatenated,

$N$28

then classified by an MLP plus softmax with cross-entropy loss. Training details reported in the cited description are AdamW with initial learning rate $N$29, weight decay $N$30, batch size $N$31, up to $N$32 epochs with early stopping on validation AUC, temperature $N$33, contrastive weight $N$34 so that $N$35, dropout $N$36 in the MLP heads, LayerNorm after each encoder block, end-to-end fine-tuning without separate pretraining, and $N$37-fold stratified cross-validation with fixed folds (Liang et al., 10 Mar 2026).

Quantitatively, the joint model improves on both single-view baselines on ADHD-200 and ABIDE. On ADHD-200, ROI-only gives $N$38 and $N$39; imaging-only gives $N$40 and $N$41; the joint model gives $N$42, $N$43, and $N$44. On ABIDE, ROI-only gives $N$45 and $N$46; imaging-only gives $N$47 and $N$48; the joint model gives $N$49, $N$50, and $N$51. The summary statement in the source is that joint projection outperforms each single-view baseline by $N$52–$N$53 in Acc/AUC (Liang et al., 10 Mar 2026).

Interpretability analyses combine Grad-CAM on the imaging branch with saliency-based attribution on the ROI branch. Imaging-only maps are described as relatively diffuse and ROI-only maps as sharp but sometimes scattered across network edges. The joint model yields spatially coherent foci in the superior frontal gyrus, precentral gyrus, orbitofrontal cortex, and hippocampal or limbic regions. This suggests that the cross-view projection is functioning less as simple feature stacking than as a geometric alignment procedure that makes complementary global and local patterns mutually usable for the downstream classifier (Liang et al., 10 Mar 2026).