DL-SH: Diverse Domain Applications

• DL-SH is a context-dependent abbreviation whose meaning is fixed by discipline, encompassing applications from diffusion MRI deep learning methods to algebraic and statistical frameworks.
• In diffusion MRI, DL-SH specifically refers to Deep SHORE, a deep learning technique that optimizes multi-shell spherical harmonic representations to improve signal reconstruction and accuracy.
• Other fields use DL-SH for specialized tasks, including supervised discrete hashing in machine learning, ExoMol line lists in spectroscopy, and invariant measure pairing in KPZ theory.

Searching arXiv for papers and relevant usages of “DL-SH” to ground the article. DL-SH is a context-dependent abbreviation rather than a single standardized term. In current arXiv usage it denotes distinct objects in diffusion MRI, molecular spectroscopy, machine learning, homotopy algebra, and KPZ probability. The most technically developed uses include a Deep SHORE generalization of single-shell deep learning in diffusion MRI, an ExoMol line list for the mercapto radical SH, supervised discrete hashing and distributional data-Shapley methods, dg/derived strong-homotopy constructions, and the directed-landscape/stationary-horizon pairing in the KPZ universality class (Nath et al., 2019, Yurchenko et al., 2018, Liu et al., 2019, Chen et al., 2018, Sorensen, 2023, Kwon et al., 2020).

1. Range of meanings

The literature indicates that DL-SH is best read as an acronym whose meaning is fixed by domain.

Domain	Meaning of DL-SH	Representative source
Diffusion MRI	Deep learning on SH generalized to Deep SHORE	(Nath et al., 2019)
Molecular spectroscopy	ExoMol SH line list for the mercapto radical	(Yurchenko et al., 2018)
Machine learning	DL-SH-style supervised discrete hashing; DShapley	(Liu et al., 2019, Kwon et al., 2020)
Homotopy algebra	dg/derived strong-homotopy Leibniz or Lie-Rinehart structures	(Chen et al., 2018, Huebschmann, 2013)
KPZ theory	Directed landscape–stationary horizon pairing	(Sorensen, 2023)

A common source of confusion is that the same string combines unrelated expansions of “DL” and “SH.” In diffusion MRI, SH denotes spherical harmonics and SHORE extends that basis. In spectroscopy and astrochemistry, SH denotes the mercapto radical or the thiol functional group. In KPZ theory, SH denotes the stationary horizon. In homotopy algebra, “sh” denotes strong homotopy.

2. Diffusion-MRI usage: from DL-SH to Deep SHORE

In diffusion MRI, DL-SH originally referred to deep learning on spherical harmonic coefficients of single-shell diffusion-weighted MRI, and Deep SHORE generalizes that paradigm to multi-shell, multi-$b$-value data by replacing purely angular SH coefficients with SHORE coefficients (Nath et al., 2019). The core representation is the SHORE expansion

$$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$

so radial dependence in $q=\|\mathbf{q}\|$ is modeled jointly with angular structure. The same framework is used to represent the observed diffusion signal and, after fitting histological fiber-orientation distributions on a fixed directional set, the target FOD.

The architectural realization is a fully connected DNN operating on coefficient vectors rather than raw voxelwise signals. The reported configuration uses 50-dimensional SHORE input at radial order 6 or 45-dimensional SH input at 8th order, hidden layers $x_1=400$, $x_2=45$, $x_3=200$, $x_4=45$, $x_5=200$, a residual connection across $x_2$, $x_3$, and $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$0, ELU nonlinearities, RMSProp optimization, and batch size 1000. A distinctive component is “learning on a data-optimized SHORE manifold”: the SHORE scale parameter $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$1 is optimized per dataset by L-BFGS using a signal-reconstruction objective before supervised training.

The paper evaluates the method on ex-vivo squirrel-monkey MRI–histology data with four shells and on in-vivo human data from two 3T scanners. For the ex-vivo task, optimized SHORE $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$2 SHORE-HFOD achieved median ACC $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$3 on single shell and $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$4 on four shells, improving over the reported single-shell baselines $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$5 and $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$6, and over the multi-shell comparison $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$7, with $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$8. In the in-vivo scanner-consistency experiment, Deep SHORE reached ACC $$E(\mathbf{q})=\sum_{n,\ell,m} c_{n\ell m}\,G_{n\ell}(q,\zeta)\,Y_\ell^m(\mathbf{u}),$$9, compared with $q=\|\mathbf{q}\|$0, $q=\|\mathbf{q}\|$1, and $q=\|\mathbf{q}\|$2 for the other multi-shell methods. This establishes DL-SHORE as a direct extension of single-shell DL-SH to varying $q=\|\mathbf{q}\|$3-values, numbers of shells, and gradient-direction counts (Nath et al., 2019).

3. Spectroscopic and astrochemical usages of SH and $q=\|\mathbf{q}\|$4SH

In molecular spectroscopy, DL-SH denotes the ExoMol ground-state line list for the mercapto radical SH. The line list covers transitions within the $q=\|\mathbf{q}\|$5 ground state and was generated for five isotopologues: $q=\|\mathbf{q}\|$6SH, $q=\|\mathbf{q}\|$7SH, $q=\|\mathbf{q}\|$8SH, $q=\|\mathbf{q}\|$9SH, and $x_1=400$0SD. For $x_1=400$1SH the line list contains 2326 rovibronic states and 81,348 transitions, with $x_1=400$2, effective vibrational coverage to $x_1=400$3, and completeness for temperatures up to 5000 K (Yurchenko et al., 2018). The construction combines MRCI+Q electronic-structure calculations with DKH4 and the AWC5Z basis, analytic EMO and morphing representations, empirical refinement to laboratory transitions, variational nuclear-motion calculations with Duo, and an ab initio diagonal $x_1=400$4–$x_1=400$5 dipole-moment curve.

This spectroscopic usage is separate from astrochemical work on thiol-bearing molecules, where “$x_1=400$6SH” denotes the thiol functional group. A recent example is the first identification of trans-dithioformic acid, t-HC(S)SH, toward NGC 1333 IRAS 4A2. The reported column density is $x_1=400$7, the excitation temperature is $x_1=400$8 K, and the fractional abundance with respect to $x_1=400$9 is $x_2=45$0. The associated UCLCHEM warm-up model attributes formation primarily to the barrierless grain-surface reaction $x_2=45$1 (Manna et al., 2024). This suggests that, in chemistry-oriented contexts, SH-related abbreviations often encode sulfur functionality rather than any deep-learning or harmonic-analysis construct.

4. Machine-learning usages: supervised hashing and distributional data valuation

One machine-learning use treats DL-SH as a family resemblance rather than a fixed acronym. In supervised discrete hashing, S2DHMLR is described as fitting squarely into DL-SH-style methods because it learns binary supervised hash codes directly under a discrete constraint while exploiting label information through mutual regression (Liu et al., 2019). Its defining choice is a single stable projection $x_2=45$2 satisfying

$x_2=45$3

instead of two separate label-to-code and code-to-label projections. The full optimization couples this mutual regression with a feature-to-code projection $x_2=45$4, and is solved by alternating updates: a Sylvester equation for $x_2=45$5, discrete cyclic coordinate descent for the binary code matrix $x_2=45$6, and ridge regression for $x_2=45$7. A boosting stage concatenates multiple runs and retains the $x_2=45$8 rows with the smallest bit-balance degree. On CIFAR-10, MS-COCO, and NUS-WIDE, the paper reports that S2DHMLR generally outperforms non-deep baselines, and that S2D-boost improves further; one example gives, on MS-COCO at 48 bits, approximate mAP values of $x_2=45$9 for FSDH, $x_3=200$0 for DHN/DSH, $x_3=200$1 for S2DHMLR, and $x_3=200$2 for S2D-boost.

A separate usage appears in data valuation, where DShapley is also glossed as DL-SH. Distributional Shapley replaces fixed-dataset Shapley values with the expected marginal contribution of a point under datasets sampled from $x_3=200$3: $x_3=200$4 For linear regression, binary classification, and KDE, the paper derives analytic expressions or bounds that reduce computation by several orders of magnitude relative to prior Monte Carlo estimators (Kwon et al., 2020). In the Gaussian linear-regression case, DShapley depends only on the Mahalanobis distance $x_3=200$5 and squared error $x_3=200$6. In the logistic case, the analysis proceeds through IRLS-transformed variables $x_3=200$7. In KDE, the leading term is the negative integrated squared error of the KDE induced by the valued point or set. This usage is statistical rather than architectural: the valued object is a datum rather than a model representation.

5. Derived and strong-homotopy algebraic meanings

In homotopy algebra, DL-SH denotes dg/derived strong-homotopy structures rather than a particular algorithm. One paper gives a functorial construction of sh Leibniz algebras from dg-module-valued derivations over a commutative dg algebra $x_3=200$8 (Chen et al., 2018). Starting from a dg derivation $x_3=200$9, with $x_4=45$0, the construction uses the $x_4=45$1-twisted Atiyah class $x_4=45$2 as the binary operation $x_4=45$3, the differential on $x_4=45$4 as $x_4=45$5, and defines higher operations recursively by covariant differentiation: $x_4=45$6 The resulting sequence $x_4=45$7 satisfies the Leibniz$x_4=45$8 identities, yielding a Kapranov Leibniz$x_4=45$9 algebra. The paper further proves that this process defines a functor from the category of dg-module-valued derivations to the category of Leibniz$x_5=200$0 $x_5=200$1-algebras, and recovers the Chen–Stiénon–Xu sh Leibniz construction for Lie pairs.

A closely related paper develops the Maurer–Cartan side of the same general theme for sh Lie-Rinehart algebras (Huebschmann, 2013). There the essential data are a filtered coalgebra perturbation $x_5=200$2 on $x_5=200$3 and a Lie algebra twisting cochain $x_5=200$4. These induce a generalized Cartan–Chevalley–Eilenberg differential on $x_5=200$5,

$x_5=200$6

producing a filtered multi derivation chain algebra. Under a reflexivity hypothesis, the multi derivation Maurer–Cartan algebra is equivalent to the sh Lie-Rinehart structure itself. In this branch of the literature, “DL-SH” is therefore best understood as shorthand for dg or derived strong-homotopy algebraic structures built from Atiyah classes, coderivations, and twisting cochains.

6. KPZ usage: directed landscape and stationary horizon

In KPZ theory, DL-SH refers to the interaction between the directed landscape $x_5=200$7 and the stationary horizon $x_5=200$8 (Sorensen, 2023). The directed landscape is the universal scaling limit of last-passage-percolation models, characterized by the metric composition law

$x_5=200$9

The stationary horizon is a random element $x_2$0 such that, for each fixed direction $x_2$1, $x_2$2 is a two-sided Brownian motion with variance $x_2$3, drift $x_2$4, and $x_2$5. Its finite-dimensional distributions are constructed from the Brownian queue map $x_2$6, and the process is monotone in $x_2$7.

The central structural result is that the stationary horizon is the unique coupled invariant distribution for the KPZ fixed point. If $x_2$8 and $x_2$9 are independent, then for every $x_3$0,

$x_3$1

where $x_3$2. A major consequence is that SH describes the Busemann process for the directed landscape. The dissertation also shows that SH governs the structure of semi-infinite geodesics in all directions and appears as the scaling limit of multi-species invariant measures for TASEP. In this context, SH means “stationary horizon,” not spherical harmonics, thiol chemistry, or strong homotopy.

Across these literatures, DL-SH is therefore not a single concept but a family of domain-specific abbreviations. Its meanings range from a SHORE-based deep representation for multi-shell diffusion MRI, to spectroscopic and astrochemical SH/$x_3$3SH notation, to discrete hashing and distributional data valuation, to derived strong-homotopy algebra, and to the stationary horizon as the central invariant measure paired with the directed landscape.