Light-Cone Feature Selection

• Light-Cone Feature Selection is a framework defining features based on causal or observational light-cone geometry in domains like quantum machine learning and cosmology.
• In quantum applications, LCFS leverages local quantum kernels and kernel-target alignment to jointly optimize circuit parameters and cone weights, enhancing feature relevance.
• In cosmological and remote-sensing contexts, LCFS informs the extraction of angular, radial, and tomographic features that respect the evolving geometry of survey data.

Searching^^^^^^^^5 arXiv5^^^^^^^^ for papers on light-cone feature selection and related formulations. {"^^^^^^^^5query5 feature selection^^^^^^^^5\5 arXiv5^^^^^^^^", "^^^^^^^^5max_results5 ^^^^5\5query5^^^^} arxiv_search(^^^^^^^^5query5 feature selection^^^^^^^^5\5 ^^^^^^^^5max_results5 Light-Cone Feature Selection (LCFS) denotes a family of feature-selection procedures in which the organizing structure is a light-cone rather than an unordered list of coordinates. In quantum machine learning, the relevant object is the past causal cone of a measured qubit or parameterized gate in a parameterized quantum circuit, and LCFS selects or ranks those cones through local quantum kernels and kernel-target alignment (&&&^^^^5query5^^^^&&&). In cosmology, the relevant object is the observational past light cone: feature construction must respect the evolving geometry of galaxy-survey data along the line of sight, and informative angular, radial, tomographic, or summary-statistic features can be ranked directly on the light cone (&&&^^^^5\5^^^^&&&). A remote-sensing application to methane hyperspectral images adopts the quantum-circuit meaning and explicitly distinguishes it from spatiotemporal past/future light-cones (&&&^^^^5 arXiv5^^^^&&&). The term therefore names a methodological pattern rather than a single domain-specific algorithm.

^^^^5\5^^^^. Terminology and conceptual scope

The literature uses “light-cone” in at least two distinct senses. In quantum machine learning, a light-cone is the minimal subset of qubits, gates, and layers that can influence a local probe in a circuit. In cosmology, the light cone is the past observational manifold parametrized by redshift and sky direction, where radial evolution breaks translational invariance and changes what constitutes an informative feature (&&&^^^^5query5^^^^&&&).

Domain	Meaning of “light-cone”	Feature object
Quantum machine learning	Past causal cone of a measurement or parameterized gate	Local kernel term or cone-defined subspace
Methane hyperspectral imaging	Same quantum-circuit causal-cone notion	Spectral band or derived index ranked through cone participation
Cosmology	Past light-cone geometry of observations	Angular multipoles, tomographic bins, summary statistics, or learned slice-wise representations

In the quantum formulation, LCFS is an embedded method: it learns importance weights jointly with the kernel rather than performing a separate wrapper-style subset search. In the cosmological formulation, the same phrase refers to selecting representations that respect line-of-sight evolution and preserve non-Gaussian, multi-scale information. A common misconception is to treat these usages as interchangeable. The methane hyperspectral study states the distinction explicitly: its “light-cone” is “analogous to causal cones in quantum circuits and tensor networks, not to be confused with the past/future light-cones in spatiotemporal dynamical systems” (&&&^^^^5 arXiv5^^^^&&&).

^^^^5 arXiv5^^^^. Quantum-machine-learning formulation

The general LCFS framework for quantum machine learning treats each light-cone as a feature and assigns it a nonnegative weight inside a decomposable kernel,

PRESERVED_PLACEHOLDER_^^^^5query5^^^^

Two concrete realizations are developed. For the projected quantum kernel (PQK),

PRESERVED_PLACEHOLDER_^^^^5\5^^^^

where PRESERVED_PLACEHOLDER_^^^^5 arXiv5^^^^ is the single-qubit reduced density matrix on qubit PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^. For the simplified quantum Fisher kernel (QFK),

PRESERVED_PLACEHOLDER_^^^^5\5^^^^

In PQK the probe is a one-qubit measurement; in QFK it is the parameterized gate carrying $\theta_l$. The cone for each term is obtained by backward traversal through the circuit graph, including additional qubits whenever a multi-qubit gate is encountered (&&&^^^^5query5^^^^&&&).

Training is organized around kernel-target alignment (KTA),

$$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$

with alternating optimization over circuit parameters $\theta$ and cone weights $\lambda$. With $\theta$ fixed, the PRESERVED_PLACEHOLDER_^^^^5\5query5^^^^ update is a convex quadratic program:

PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^

where

PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5^^^^

The implementation described in the paper uses Adam with learning rate PRESERVED_PLACEHOLDER_^^^^5\5max_results5^^^^ for about PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^ iterations per outer loop, and solves the quadratic program with CVXOPT (&&&^^^^5query5^^^^&&&).

For classical inputs, cone weights can be mapped back to coordinate-level importance scores. If PRESERVED_PLACEHOLDER_^^^^5\55^^^^ counts how often classical feature PRESERVED_PLACEHOLDER_^^^^5\5^^^^6 appears in cone PRESERVED_PLACEHOLDER_^^^^5\5^^^^7, then

PRESERVED_PLACEHOLDER_^^^^5\5^^^^8

This compensates for the fact that deeper cones cover more coordinates and therefore would otherwise inflate apparent relevance.

The empirical demonstrations in the foundational LCFS study show four uses of the framework. On the parityPRESERVED_PLACEHOLDER_^^^^5\5^^^^9 task with PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5^^^^ and depth PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5^^^^, QFK reaches PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5^^^^ and cleanly selects PRESERVED_PLACEHOLDER_^^^^5 arXiv5max_results5^^^^, whereas PQK reaches PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5^^^^ and also selects extraneous PRESERVED_PLACEHOLDER_^^^^5 arXiv55^^^^ because of cone growth. On parityPRESERVED_PLACEHOLDER_^^^^5 arXiv5^^^^6, LCFS-guided feature-to-qubit reordering changes poor or negative test KTA without reordering—PRESERVED_PLACEHOLDER_^^^^5 arXiv5^^^^7 for QFK and PRESERVED_PLACEHOLDER_^^^^5 arXiv5^^^^8 for PQK—into PRESERVED_PLACEHOLDER_^^^^5 arXiv5^^^^9 and PRESERVED_PLACEHOLDER_^^^^5max_results5query5^^^^, respectively. On the Breast Cancer dataset, QFK test KTA improves from PRESERVED_PLACEHOLDER_^^^^5max_results5\5^^^^ to PRESERVED_PLACEHOLDER_^^^^5max_results5 arXiv5^^^^ and PQK from PRESERVED_PLACEHOLDER_^^^^5max_results5max_results5^^^^ to PRESERVED_PLACEHOLDER_^^^^5max_results5\5^^^^ after reordering, while yielding sparser importance profiles. For compression, keeping only the largest-PRESERVED_PLACEHOLDER_^^^^5max_results55^^^^ cone on parityPRESERVED_PLACEHOLDER_^^^^5max_results5^^^^6 reduces QFK KTA from PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^7 to PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^8 and PQK from PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^9 to PRESERVED_PLACEHOLDER_^^^^5\5query5^^^^. The same formalism also performs subspace selection for quantum data, where no classical coordinate system is available (&&&^^^^5query5^^^^&&&).

These results establish the central interpretation of LCFS in QML: the “feature” is a causal subcircuit or subspace, not merely an input coordinate. This suggests that LCFS is simultaneously a feature selector, an architecture-search heuristic, and a pruning rule.

^^^^5max_results5^^^^. Methane hyperspectral imaging and quantum-kernel LCFS

A concrete application of LCFS appears in methane detection and localization from hyperspectral images collected by AVIRIS-NG over geographically diverse fossil-fuel sites. The data come from the STARCOP dataset. Eight bands are used for analysis—PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^, PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5^^^^, PRESERVED_PLACEHOLDER_^^^^5\5max_results5^^^^, PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^, PRESERVED_PLACEHOLDER_^^^^5\55^^^^, PRESERVED_PLACEHOLDER_^^^^5\5^^^^6, PRESERVED_PLACEHOLDER_^^^^5\5^^^^7, and PRESERVED_PLACEHOLDER_^^^^5\5^^^^8 nm—and the last three lie in the shortwave infrared region containing strong PRESERVED_PLACEHOLDER_^^^^5\5^^^^9 absorption features near $\theta_l$^^^^5query5^^^^–θl\theta_l^^^^5\5^^^^ (&&&^^^^5 arXiv5^^^^&&&).

The preprocessing pipeline is explicitly superpixel-based. Extended SLIC is computed over all available bands, band values are averaged within each superpixel, a mag^^^^5\5^^^^c methane enhancement map is produced and averaged per superpixel, and the binary plume mask assigns labels by majority within each superpixel. The dataset comprises $\theta_l$^^^^5 arXiv5^^^^ hyperspectral images of size $\theta_l$^^^^5max_results5^^^^ for the selected bands. Training uses only $\theta_l$^^^^5\5^^^^ labeled superpixels—$\theta_l$5 methane and $\theta_l$6 background—because the kernels are computed in simulation; the test set $\theta_l$7 contains all methane superpixels plus randomly sampled background superpixels, with $\theta_l$8 (&&&^^^^5 arXiv5^^^^&&&).

The quantum model uses $\theta_l$9 qubits and $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$^^^^5query5^^^^ alternating layers of one-qubit data-reupload gates $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$^^^^5\5^^^^ and two-qubit entanglers $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$^^^^5 arXiv5^^^^. Measuring each qubit yields a local kernel

$$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$^^^^5max_results5^^^^

and the global kernel is

$$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$^^^^5\5^^^^

where $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$5 is set by centered alignment and normalized so that $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$6. Feature importance is defined through light-cone participation counts $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$7 of feature $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$8 in the past cone of measured qubit $$\mathrm{KTA}(K,Y)=\frac{\langle K,Y\rangle_F}{\|K\|_F\|Y\|_F} =\frac{\sum_{p,q} y_p y_q k(x_p,x_q)} {\sqrt{\sum_{p,q} k(x_p,x_q)^2}\sqrt{\sum_{p,q}(y_p y_q)^2}},$$9. The paper displays

$\theta$^^^^5query5^^^^

but also states that features with high values of $\theta$^^^^5\5^^^^ are those “re-uploaded many times in the most influential local kernels.” The text therefore identifies a mismatch between the displayed inverse formula and the intended monotonic behavior (&&&^^^^5 arXiv5^^^^&&&).

Three SVM baselines are compared: linear, RBF, and quantum-kernel SVM. Without mag^^^^5\5^^^^c, the reported metrics on $\theta$^^^^5 arXiv5^^^^ are: for SVML, accuracy $\theta$^^^^5max_results5^^^^, sensitivity $\theta$^^^^5\5^^^^, specificity $\theta$5, F-score $\theta$6, MCC $\theta$7; for SVMRBF, accuracy $\theta$8, sensitivity $\theta$9, specificity $\lambda$^^^^5query5^^^^, F-score $\lambda$^^^^5\5^^^^, MCC $\lambda$^^^^5 arXiv5^^^^; for SVMQ, accuracy $\lambda$^^^^5max_results5^^^^, sensitivity $\lambda$^^^^5\5^^^^, specificity $\lambda$5, F-score $\lambda$6, MCC $\lambda$7 (&&&^^^^5 arXiv5^^^^&&&).

When mag^^^^5\5^^^^c is inserted under a fixed $\lambda$8 qubit budget by dropping one band, two quantum configurations stand out. Dropping band $\lambda$9 ($\theta$^^^^5query5^^^^ nm) yields accuracy $\theta$^^^^5\5^^^^, sensitivity $\theta$^^^^5 arXiv5^^^^, specificity $\theta$^^^^5max_results5^^^^, F-score $\theta$^^^^5\5^^^^, MCC $\theta$5, and $\theta$6, which the paper interprets as no significant difference versus ground truth. Dropping band $\theta$7 ($\theta$8 nm) yields accuracy $\theta$9, sensitivity PRESERVED_PLACEHOLDER_^^^^5\5query5query5^^^^, specificity PRESERVED_PLACEHOLDER_^^^^5\5query5\5^^^^, F-score PRESERVED_PLACEHOLDER_^^^^5\5query5 arXiv5^^^^, MCC PRESERVED_PLACEHOLDER_^^^^5\5query5max_results5^^^^, and PRESERVED_PLACEHOLDER_^^^^5\5query5\5^^^^ (&&&^^^^5 arXiv5^^^^&&&).

The ablation results show that feature ranking is not fully stable. The top three mean importance scores are bands PRESERVED_PLACEHOLDER_^^^^5\5query55^^^^ (PRESERVED_PLACEHOLDER_^^^^5\5query5^^^^6), PRESERVED_PLACEHOLDER_^^^^5\5query5^^^^7 (PRESERVED_PLACEHOLDER_^^^^5\5query5^^^^8), and PRESERVED_PLACEHOLDER_^^^^5\5query5^^^^9 (PRESERVED_PLACEHOLDER_^^^^5\5\5query5^^^^), whereas the top three mean sums of metrics are PRESERVED_PLACEHOLDER_^^^^5\5\5\5^^^^ (PRESERVED_PLACEHOLDER_^^^^5\5\5 arXiv5^^^^), PRESERVED_PLACEHOLDER_^^^^5\5\5max_results5^^^^ (PRESERVED_PLACEHOLDER_^^^^5\5\5\5^^^^), and PRESERVED_PLACEHOLDER_^^^^5\5\55^^^^ (PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^6). The paper states that both measures have relatively large standard deviations and agree only on the importance of feature PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^7 (&&&^^^^5 arXiv5^^^^&&&). A plausible implication is that LCFS in this setting is sensitive to circuit topology and small-sample variability, even when it improves downstream metrics by enabling mag^^^^5\5^^^^c inclusion.

^^^^5\5^^^^. Cosmological light-cone geometry and feature construction

In cosmology, the light cone is the natural data domain of galaxy surveys. Observations are indexed by PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^8, and an observable is modeled as

PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^9

Because the radial coordinate mixes space and time, translational invariance is broken along the line of sight, while angular directions retain statistical isotropy. This is why spherical harmonics and spherical Fourier–Bessel (SFB) decompositions are natural, and why feature-selection rules derived from cubic-box intuition can mis-rank informative modes in wide or deep surveys (&&&^^^^5\5\5^^^^&&&).

The SFB decomposition writes

PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5query5^^^^

with covariance

PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5\5^^^^

The radial spectrum PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5 arXiv5^^^^ is generally non-diagonal because light-cone evolution mixes radial modes. For angular or tomographic data, the corresponding Fisher matrix is

PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5max_results5^^^^

The same framework yields per-PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5\5^^^^ and per-bin information measures for selecting observables, multipole ranges, cross-spectra, or tomographic bins directly on the past light cone (&&&^^^^5\5\5^^^^&&&).

The more recent machine-learning study translates this geometric constraint into representation design. It uses AbacusSummit halo lightcone mocks from a corner of the main box plus two periodic copies, selecting PRESERVED_PLACEHOLDER_^^^^5\5 arXiv55^^^^ and PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5^^^^6, with halo selection PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5^^^^7 and number densities PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5^^^^8–PRESERVED_PLACEHOLDER_^^^^5\5 arXiv5^^^^9. Redshift-space distortions are included through

PRESERVED_PLACEHOLDER_^^^^5\5max_results5query5^^^^

and an Alcock–Paczynski mapping to a fiducial cosmology is applied:

PRESERVED_PLACEHOLDER_^^^^5\5max_results5\5^^^^

The key methodological claim is that a ^^^^5max_results5^^^^D CNN on a single gridded lightcone implicitly imposes translational invariance along the line of sight and can therefore mix features from different lookback times or learn spurious invariances (&&&^^^^5\5^^^^&&&).

This motivates a geometry-aware “slice–project–analyze” strategy: divide the light cone into thin redshift slices, project each slice to a HEALPix sphere, and then analyze the stack with a ^^^^5 arXiv5^^^^D CNN. In the reported implementation, nine slices cover PRESERVED_PLACEHOLDER_^^^^5\5max_results5 arXiv5^^^^, HEALPix uses PRESERVED_PLACEHOLDER_^^^^5\5max_results5max_results5^^^^, each slice is mapped to a PRESERVED_PLACEHOLDER_^^^^5\5max_results5\5^^^^ Cartesian patch over PRESERVED_PLACEHOLDER_^^^^5\5max_results55^^^^, and the resulting PRESERVED_PLACEHOLDER_^^^^5\5max_results5^^^^6 tensor is processed jointly (&&&^^^^5\5^^^^&&&).

5. Summary statistics, learned features, and comparative performance in cosmology

The cosmological comparison includes four feature families: slice-wise image features processed by a ^^^^5 arXiv5^^^^D CNN, spherical harmonic coefficients PRESERVED_PLACEHOLDER_^^^^5\5max_results5^^^^7, wavelet scattering transform (WST) coefficients, and the angular two-point correlation function PRESERVED_PLACEHOLDER_^^^^5\5max_results5^^^^8, all evaluated on AbacusSummit halo lightcones (&&&^^^^5\5^^^^&&&).

For the harmonic representation, each field is expanded as

PRESERVED_PLACEHOLDER_^^^^5\5max_results5^^^^9

with angular power spectrum

PRESERVED_PLACEHOLDER_^^^^5\5\5query5^^^^

and

PRESERVED_PLACEHOLDER_^^^^5\5\5\5^^^^

In practice, with PRESERVED_PLACEHOLDER_^^^^5\5\5 arXiv5^^^^ and PRESERVED_PLACEHOLDER_^^^^5\5\5max_results5^^^^, the PRESERVED_PLACEHOLDER_^^^^5\5\5\5^^^^ are compressed by PCA to PRESERVED_PLACEHOLDER_^^^^5\5\55^^^^ components per slice explaining PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^6 of the variance. For WST, the full lightcone density is gridded into PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^7, tiled into PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^8 sub-regions of size PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^9, and each sub-volume yields PRESERVED_PLACEHOLDER_^^^^5\55query5^^^^ coefficients for PRESERVED_PLACEHOLDER_^^^^5\55\5^^^^ scales and PRESERVED_PLACEHOLDER_^^^^5\55 arXiv5^^^^ orientations; PCA reduces each to PRESERVED_PLACEHOLDER_^^^^5\55max_results5^^^^ components explaining PRESERVED_PLACEHOLDER_^^^^5\55\5^^^^ of the variance. For the ^^^^5 arXiv5^^^^PCF, the estimator is Landy–Szalay,

PRESERVED_PLACEHOLDER_^^^^5\555^^^^

with PRESERVED_PLACEHOLDER_^^^^5\55^^^^6 linear bins in PRESERVED_PLACEHOLDER_^^^^5\55^^^^7 per slice and jackknife covariance over PRESERVED_PLACEHOLDER_^^^^5\55^^^^8 subregions per slice (&&&^^^^5\5^^^^&&&).

The CNN+^^^^5 arXiv5^^^^D architecture has four convolutional blocks with filters PRESERVED_PLACEHOLDER_^^^^5\55^^^^9, PRESERVED_PLACEHOLDER_^^^^5\5^^^^6^^^^5query5^^^^ convolutions, batch normalization, ReLU, and PRESERVED_PLACEHOLDER_^^^^5\5^^^^6^^^^5\5^^^^ max-pooling, followed by adaptive average pooling to PRESERVED_PLACEHOLDER_^^^^5\5^^^^6^^^^5 arXiv5^^^^, flattening to PRESERVED_PLACEHOLDER_^^^^5\5^^^^6^^^^5max_results5^^^^ features, and a dense regressor with DensePRESERVED_PLACEHOLDER_^^^^5\5^^^^6^^^^5\5 ReLU, DropoutPRESERVED_PLACEHOLDER_^^^^5\5^^^^65, and OutputPRESERVED_PLACEHOLDER_^^^^5\5^^^^66 for PRESERVED_PLACEHOLDER_^^^^5\5^^^^67. FC models for PRESERVED_PLACEHOLDER_^^^^5\5^^^^68, WST, and ^^^^5 arXiv5^^^^PCF use the same DensePRESERVED_PLACEHOLDER_^^^^5\5^^^^69 + ReLU + DropoutPRESERVED_PLACEHOLDER_^^^^5\5^^^^7^^^^5query5^^^^ + OutputPRESERVED_PLACEHOLDER_^^^^5\5^^^^7^^^^5\5^^^^ head. Training uses Adam with learning rate PRESERVED_PLACEHOLDER_^^^^5\5^^^^7^^^^5 arXiv5^^^^, batch size PRESERVED_PLACEHOLDER_^^^^5\5^^^^7^^^^5max_results5 and mean-squared error loss (&&&^^^^5\5^^^^&&&).

The reported test losses are:

• FC+WST: PRESERVED_PLACEHOLDER_^^^^5\5^^^^7^^^^5\5
• FC+PRESERVED_PLACEHOLDER_^^^^5\5^^^^75: PRESERVED_PLACEHOLDER_^^^^5\5^^^^76
• CNN+^^^^5 arXiv5^^^^D: PRESERVED_PLACEHOLDER_^^^^5\5^^^^77
• FC+^^^^5 arXiv5^^^^PCF: PRESERVED_PLACEHOLDER_^^^^5\5^^^^78

Sample parameter-level metrics illustrate the ranking. For PRESERVED_PLACEHOLDER_^^^^5\5^^^^79, CNN+^^^^5 arXiv5^^^^D gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^8^^^^5query5 RMSE PRESERVED_PLACEHOLDER_^^^^5\5^^^^8^^^^5\5 FC+WST gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^8^^^^5 arXiv5^^^^, PRESERVED_PLACEHOLDER_^^^^5\5^^^^8^^^^5max_results5 FC+PRESERVED_PLACEHOLDER_^^^^5\5^^^^8^^^^5\5^^^^ gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^85, PRESERVED_PLACEHOLDER_^^^^5\5^^^^86; FC+^^^^5 arXiv5^^^^PCF gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^87, PRESERVED_PLACEHOLDER_^^^^5\5^^^^88. For PRESERVED_PLACEHOLDER_^^^^5\5^^^^89, CNN+^^^^5 arXiv5^^^^D gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^9^^^^5query5 RMSE PRESERVED_PLACEHOLDER_^^^^5\5^^^^9^^^^5\5 while FC+WST gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^9^^^^5 arXiv5^^^^, PRESERVED_PLACEHOLDER_^^^^5\5^^^^9^^^^5max_results5^^^^ and FC+^^^^5 arXiv5^^^^PCF gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^9^^^^5\5 For PRESERVED_PLACEHOLDER_^^^^5\5^^^^95, CNN+^^^^5 arXiv5^^^^D gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^96, RMSE PRESERVED_PLACEHOLDER_^^^^5\5^^^^97, FC+WST gives PRESERVED_PLACEHOLDER_^^^^5\5^^^^98, PRESERVED_PLACEHOLDER_^^^^5\5^^^^99, and FC+^^^^5 arXiv5^^^^PCF gives PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5query5^^^^, PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5\5^^^^ (&&&^^^^5\5^^^^&&&).

The fiducial multiple-realization experiment modifies the ranking. For cosmology c^^^^5query5query5query5^^^^ with PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5 arXiv5^^^^ realizations, CNN+^^^^5 arXiv5^^^^D yields PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5max_results5^^^^, PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5\5^^^^, PRESERVED_PLACEHOLDER_^^^^5 arXiv5query55^^^^, PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5^^^^6, PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5^^^^7, and PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5^^^^8. FC+WST remains competitive and is tighter for some parameters, for example PRESERVED_PLACEHOLDER_^^^^5 arXiv5query5^^^^9, but the paper states that CNN+^^^^5 arXiv5^^^^D attains the smallest statistical uncertainties overall across the fiducial ensemble (&&&^^^^5\5^^^^&&&).

The qualitative interpretation is explicit. WST outperforms ^^^^5 arXiv5^^^^PCF-based methods because it captures higher-order, multi-scale statistics through cascaded wavelet moduli and low-pass averages, including filamentary anisotropy and halo/void morphology across scales. Appendix-level comparisons show larger inter-cosmology differences at high multipoles in PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5query5^^^^ and at small angles in PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5\5^^^^, while WST coefficients vary with scale PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5 arXiv5^^^^ and orientation PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5max_results5^^^^, reflecting multi-scale anisotropy. These regimes contribute strongly to constraints on PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5\5^^^^ and PRESERVED_PLACEHOLDER_^^^^5 arXiv5\55^^^^, and PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5^^^^6 benefits from broad-scale sensitivity in both WST and high-PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5^^^^7 PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5^^^^8 (&&&^^^^5\5^^^^&&&).

6. Limitations, ambiguities, and future directions

Across the literature, LCFS is limited by the geometry it assumes and by the objects it treats as features. In PQK, cone growth with depth can create false positives by sweeping many classical coordinates into a measurement cone; this is why the QFK formulation is described as sharper for precise localization on the parity tasks (&&&^^^^5query5^^^^&&&). In the methane study, feature-importance scores have relatively large standard deviations, centered-alignment optimization of PRESERVED_PLACEHOLDER_^^^^5 arXiv5\5^^^^9 is described as problematic, and the training set is restricted to PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5query5^^^^ samples because kernel computation is simulator-limited (&&&^^^^5 arXiv5^^^^&&&). These facts indicate that topology dependence, training-sample scarcity, and simulator cost remain core constraints for quantum LCFS.

In cosmology, limitations arise from both modeling and representation. The CNN+^^^^5 arXiv5^^^^D pipeline assumes approximate stationarity within each redshift slice and does not explicitly model temporal dynamics along the line of sight. The PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5\5^^^^ and ^^^^5 arXiv5^^^^PCF summaries are fundamentally two-point descriptions; under complex survey masks, pseudo-PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5 arXiv5^^^^ with mode-coupling matrices is required to debias PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5max_results5^^^^ estimates. WST sensitivity depends on configuration choices such as wavelet family, PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5\5^^^^, PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv55^^^^, and PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5^^^^6, and the reported implementation uses a ^^^^5max_results5^^^^D Euclidean gridding rather than a spherical scattering construction. More generally, supervised learning depends on the realism of mocks and halo-occupation prescriptions, which motivates domain adaptation, nuisance marginalization, and cross-survey validation (&&&^^^^5\5^^^^&&&).

Future directions are correspondingly domain-specific. For quantum LCFS, the literature points to more robust alignment optimization, regularization for PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5^^^^7 and PRESERVED_PLACEHOLDER_^^^^5 arXiv5 arXiv5^^^^8, systematic cone design and topology search, larger training sets, and extension to other Earth-observation tasks or other quantum-data settings (&&&^^^^5 arXiv5^^^^&&&). For cosmology, the reported directions include spatiotemporal CNNs or transformers across slices, spherical CNNs on HEALPix, learned scattering networks, hybrid CNN+WST models, emulator-based WST likelihoods, and hybrid machine-learning-plus-likelihood pipelines (&&&^^^^5\5^^^^&&&). The broader Fisher-based light-cone program also emphasizes multi-tracer combinations, cross-correlations, and explicit treatment of angular versus radial information as the route to overcoming cosmic-variance ceilings on the past light cone (&&&^^^^5\5\5^^^^&&&).

Taken together, the literature supports a narrow but technically consistent characterization: LCFS is effective when relevance is localized by a causal or observational geometry, and when the feature map preserves that geometry rather than erasing it through inappropriate invariances. In QML, the relevant objects are cone-defined subspaces and local kernels. In cosmology, they are light-cone-consistent angular, radial, tomographic, or learned summaries. The unifying principle is that feature relevance is determined by what can physically or causally influence the target observable on the chosen light-cone.