---
title: Heracles in Astronomy, ML, and Robotics
url: https://www.emergentmind.com/topics/heracles
type: topic
---

# Heracles in Astronomy, ML, and Robotics

Searching arXiv for relevant papers on “Heracles” to ground the article and disambiguate the multiple uses of the term.
Found multiple distinct research usages of “Heracles,” including Milky Way archaeology, machine learning, and humanoid control. I’ll rely on the specific papers supplied in the data block and cite them precisely.
Heracles is a name used for several unrelated research objects in contemporary literature. In Galactic archaeology, it denotes a chemically defined inner-Galaxy proto-Galactic component identified in APOGEE–Gaia analyses and modeled as a significant fragment of the proto-Milky Way [2410.16374]. In machine learning, it denotes a hybrid SSM–Transformer architecture for high-resolution image and time-series analysis built from a Hartley-kernel global SSM, a localized convolutional SSM, and attention-based token interaction [2403.18063]. In humanoid robotics, it denotes a state-conditioned diffusion middleware that operates between reference motions and low-level physics trackers, preserving identity-like tracking near nominal states and synthesizing recovery trajectories under large deviations [2603.27756]. The shared name therefore functions as a disciplinary homonym rather than a single concept.

## 1. Disciplinary scope and nomenclature

In the Milky Way literature, “Heracles” appears as a stellar structure or proto-Galactic fragment inferred from chemistry, spatial concentration, and orbital properties. Horta & Schiavon identify populations in the inner Galaxy largely associated with the Heracles structure through a purely chemical dissection of APOGEE–Gaia stellar populations, and argue that the proto-Milky Way is at least comprised of two significant fragments: the main *in situ* progenitor and the Heracles structure [2410.16374].

In machine learning, “Heracles” names a model architecture introduced for vision and time-series analysis. It is explicitly described as “a Hybrid SSM–Transformer Model for High-Resolution Image and Time-Series Analysis,” with a three-component block comprising a global SSM, a local SSM, and an attention module [2403.18063].

In humanoid control, “Heracles” names a control framework subtitled “Bridging Precise Tracking and Generative Synthesis for General Humanoid Control.” Its core contribution is a state-conditioned diffusion middleware that adapts between precise tracking and generative recovery without explicit mode switching [2603.27756].

This multiplicity of meanings is important because the astronomical, machine-learning, and robotics usages are technically unrelated, despite sharing the same label.

## 2. Heracles in Galactic archaeology: identification and chemistry

Heracles was initially characterized in APOGEE–Gaia work as a metal-poor, low-energy, high-eccentricity substructure confined to the inner few kpc of the Milky Way halo. In APOGEE DR17 + Gaia EDR3, 300 Heracles candidate stars were selected solely by orbital parameters plus a chemical cut in the [Mg/Mn]–[Al/Fe] plane. The full criteria were \(e>0.6\), \(-2.6 < E < -2.0 \times 10^5\,\mathrm{km}^2\,\mathrm{s}^{-2}\), a two-segment selection in the [Al/Fe] vs. [Mg/Mn] plane, and \([\mathrm{Fe/H}]>-1.7\); the resulting sample lay at \(R_{\rm GC}\lesssim 4\) kpc [2204.04233].

A later APOGEE–Gaia analysis used a purely chemical dissection. In that framework, stars of the “unevolved” locus in the [Mg/Mn]–[Al/Fe] plane satisfy \([\mathrm{Mg}/\mathrm{Mn}]>0.15\) or \([\mathrm{Al}/\mathrm{Fe}]<-0.2\), with boundary
\[
[\mathrm{Mg}/\mathrm{Mn}] = 2\,[\mathrm{Al}/\mathrm{Fe}] + 0.6,
\]
and Heracles is then isolated by additional cuts in the [Mg/Fe]–[Fe/H] plane, selecting the high-[Mg/Fe], moderate-[Fe/H] locus that is spatially confined to the inner Galaxy [2410.16374].

Chemically, Heracles is consistently described as metal poor and \(\alpha\)-enhanced. One APOGEE-based characterization gives a metallicity distribution spanning roughly \(-1.7 \lesssim [\mathrm{Fe/H}] \lesssim -1.0\) with a peak around \([\mathrm{Fe/H}] \approx -1.3\), median \(\langle[\mathrm{Fe/H}]\rangle \approx -1.30\) dex, and dispersion \(\sigma_{[\mathrm{Fe/H}]}\approx 0.20\) dex. For Mg, the sequence is approximately \([\mathrm{Mg}/\mathrm{Fe}] \approx +0.32\pm0.02\) dex over \(-1.7<[\mathrm{Fe/H}]<-1.2\), with no statistically significant knee detected up to \([\mathrm{Fe/H}] \sim -1.0\). Fe-peak and neutron-capture tracers were reported as \([\mathrm{Ni}/\mathrm{Fe}]\approx -0.05\) dex at \([\mathrm{Fe/H}]=-1.7\) rising to \(\sim+0.05\) dex at \(-1.1\), \([\mathrm{Mn}/\mathrm{Fe}]\) flat at \(-0.2\lesssim[\mathrm{Mn}/\mathrm{Fe}]\lesssim0.0\), and \([\mathrm{Ce}/\mathrm{Fe}]\approx -0.1\pm0.04\) at \([\mathrm{Fe/H}]\approx -1.3\) [2204.04233].

The chemically dissected proto-Galaxy analysis gives a closely related but not identical description: Heracles stars span roughly \(-1.6 \lesssim [\mathrm{Fe/H}] \lesssim -0.7\), peak around \([\mathrm{Fe/H}] \approx -1.2\), and are distinguished at fixed \([\mathrm{Fe/H}]\) by relatively high \(\alpha\)-enhancement, \([\mathrm{Mg}/\mathrm{Fe}] \sim 0.30\)–\(0.45\), compared with lower-[Mg/Fe] debris such as Gaia-Enceladus/Sausage [2410.16374].

## 3. Spatial structure, density modeling, and mass of the Galactic Heracles

The most explicit density model for Heracles is a triaxial, oblate Plummer profile. For Heracles-dominated chemical cells, the three-dimensional density is written as
\[
\nu_*(r_e)\propto \Bigl(1+\frac{r_e^2}{a^2}\Bigr)^{-5/2},
\qquad
r_e^2 = X^2 + \Bigl(\frac{Y}{p}\Bigr)^2 + \Bigl(\frac{Z}{q}\Bigr)^2,
\]
or, in mass-normalized form,
\[
\rho(r_e)=\frac{3M}{4\pi a^3}\,\Bigl(1+\frac{r_e^2}{a^2}\Bigr)^{-5/2}.
\]
For the core Heracles cell, the best-fit parameters are \(a = 3.11 \pm 0.10\;\mathrm{kpc}\), \(p = 0.77 \pm 0.03\), and \(q = 0.63 \pm 0.03\) [2410.16374].

These values are consistent with the broader proto-Galaxy fit, which finds the chemically defined populations to be well represented by a Plummer model with a scale radius of \(a\sim3.5\) kpc and an oblate ellipsoid with flattening parameters \(p\sim0.8\) and \(q\sim0.6\). The interpretation offered there is that the Milky Way plausibly hosts a low-mass, metal-poor, bulge component [2410.16374].

Mass estimates depend on the adopted chemical cell definition. Integrating the density of Cell 2 within \(r<10\,\mathrm{kpc}\) yields
\[
M_*(\mathrm{Heracles},\,r<10\,\mathrm{kpc}) = (7.2 \pm 0.2)\times10^8\,M_\odot.
\]
Summing over all three inner-Galaxy chemical cells gives a combined mass of \(\sim 9.5\times10^8\,M_\odot\), while modeling all three together gives \(9.1\pm0.2\times10^8\,M_\odot\) [2410.16374].

Morphologically, the best-fit \(q\sim0.6<1\) indicates a distinctly oblate ellipsoid, and \(p\sim0.8\) indicates only mild triaxiality. Heracles occupies the inner \(\sim5\)–\(12\) kpc apocenter halo, overlaps the classical bulge/bar region, and remains chemically distinct from the bar’s younger, higher-[Fe/H] populations. Its centrally concentrated Plummer core with \(a\sim3\) kpc was described as resembling a low-mass, metal-poor bulge component, a “poor old heart,” embedded within the boxy/peanut bar [2410.16374].

An earlier APOGEE-based interpretation, using orbital decay and chemical arguments, estimated a progenitor stellar mass \(M_\star\sim3\times10^8\,M_\odot\), with accretion at high redshift \(z\gtrsim2\), and interpreted Heracles as a building block whose debris now dominate the very inner halo [2204.04233]. This suggests that published mass and origin estimates depend sensitively on the adopted selection and modeling strategy.

## 4. Competing interpretations: accreted fragment, proto-Galactic component, or Aurora analogue

The principal controversy surrounding Galactic Heracles concerns whether it is best understood as an *ex situ* merger remnant, a distinct proto-Galactic fragment, or the chemical twin of an *in situ* population called Aurora.

One APOGEE-based study states that Heracles differs chemically from *in situ* populations such as Aurora and its inner halo counterparts in a statistically significant way. Using a 13-element \(\chi^2\) comparison,
\[
\chi^2=\sum_{i=1}^{N_{\rm el}}
\frac{\bigl([X/Fe]_{i,\rm Her}-[X/Fe]_{i,\rm ref}\bigr)^2}
{\sigma_{i,\rm Her}^2+\sigma_{i,\rm ref}^2},
\]
with \(N_{\rm el}=13\) for the inner high-\(\alpha\) comparison and \(N_{\rm el}=11\) for Heracles versus Aurora, it reports at \([\mathrm{Fe/H}]=-1.0\):
\[
\chi^2(\text{Heracles vs.\ inner high-}\alpha)=55.9,\quad p_{\chi^2}\simeq0,
\]
and
\[
\chi^2(\text{Heracles vs.\ Aurora})=47.7,\quad p\approx0.
\]
The same work argues that in every major element, especially O, Mg, and Si, Heracles is \(\sim0.08\)–\(0.12\) dex lower than the *in situ* high-\(\alpha\) sequence at the same \([\mathrm{Fe/H}]\), and interprets this as evidence that the star formation rate was lower in Heracles than in the early Milky Way [2204.04233].

A different analysis reaches the opposite conclusion. In an unsupervised decomposition of the local stellar halo, Myeong et al. report that “Aurora is entirely consistent with the chemical properties of the so-called Heracles merger.” In APOGEE, Aurora has \(\langle[\mathrm{Fe/H}]\rangle=-1.19\pm0.15\), \(\langle[\alpha/\mathrm{Fe}]\rangle=+0.32\pm0.05\), and \(\langle[\mathrm{Al}/\mathrm{Fe}]\rangle=+0.13\pm0.19\); in GALAH it has \(\langle[\mathrm{Fe/H}]\rangle=-1.07\pm0.10\), \(\langle[\alpha/\mathrm{Fe}]\rangle=+0.24\pm0.06\), \(\langle[\mathrm{Al}/\mathrm{Fe}]\rangle=+0.10\pm0.16\), together with elevated Ba, Y, and Eu. That study proposes that the Heracles signature is “the fossil relic of the earliest, bursty phase of the Milky Way’s own disk growth (‘in situ’), rather than a disrupted satellite” [2206.07744].

The later APOGEE–Gaia proto-Galaxy modeling of Horta & Schiavon again treats Heracles as one of at least two significant proto-Galactic fragments, alongside the main *in situ* progenitor. In that interpretation, Heracles is “comparably massive” to the main progenitor, formed more slowly, and contributes significantly to the metal-poor bulge/inner halo [2410.16374].

Taken together, the literature does not offer a single settled interpretation. The stable points of agreement are that Heracles is chemically old, metal poor, centrally concentrated, and associated with the ancient inner Galaxy; the disputed point is whether those properties are best attributed to an accreted building block, a separate proto-Galactic fragment, or the earliest *in situ* Milky Way starburst.

## 5. Heracles as a hybrid SSM–Transformer model for image and time-series analysis

In machine learning, Heracles is a model architecture designed to address limitations attributed both to vision transformers and to earlier vision-oriented state space models. The architecture is defined as a three-component block: a global SSM based on a real-valued Hartley kernel, a localized convolutional SSM for fine spatial detail, and an attention-based token interaction module in deeper layers. Starting from the continuous-time state-space model
\[
x'(t)=A x(t)+B u(t),\qquad y(t)=C x(t)+D u(t),
\]
the model uses zero-order-hold discretization,
\[
x_k=\bar A x_{k-1}+\bar B u_k,\qquad y_k=\bar C x_k,
\]
and the convolutional kernel view
\[
y_k=[\bar C\bar A^0\bar B,\;\bar C\bar A^1\bar B,\ldots] * u.
\]
Its global operator replaces a complex FFT by the real Hartley transform \(\mathcal{H}\), defining
\[
(\mathcal{K}v)(x)=\mathcal{H}^{-1}[R_H\cdot \mathcal{H}v](x),
\]
where \(R_H\in\mathbb{R}^{d\times d}\) is a learned Hartley-domain filter. The local stream is a discrete convolution
\[
(\mathcal{C}f)(x)=\sum_{i,j} k_{ij} f(x-z_{ij}),
\]
and deeper layers append multi-head self-attention with
\[
Q=XW_Q,\quad K=XW_K,\quad V=XW_V,\qquad
\mathrm{Attention}(X)=\mathrm{softmax}(QK^\top/\sqrt d)\,V.
\]
The layer flow is: input tokens \(X\) of shape \((N,d)\); split into global and local streams; sum streams, apply LayerNorm and a two-layer MLP; use only SSM+MLP in the first \(\alpha\) blocks and append multi-head attention in the later \(L-\alpha\) stages [2403.18063].

For ImageNet-1K, the training recipe is AdamW with \((\beta_1=0.9,\beta_2=0.999)\), \(lr=1e{-5}\), \(weight\_decay=0.05\), a 10-epoch linear warm-up plus 310-epoch cosine decay, batch size 128 on \(8\times\)V100 GPUs, \(224\times224\) input resolution, and RandAug, CutMix, MixToken, and Token Labeling [2403.18063].

| Variant | Params / FLOPs | ImageNet top-1 |
|---|---:|---:|
| Heracles-C-Small | 21.7 M / 4.1 G | 84.5% |
| Heracles-C-Base | 32.5 M / 6.5 G | 85.2% |
| Heracles-C-Large | 54.1 M / 13.4 G | 85.9% |
| Heracles-C-Huge | 156.7 M / 39.3 G | 86.4% |

The reported ImageNet comparisons state that Heracles-C-Small at \(4.1\) G achieves \(84.5\%\) versus BiFormer-S \(84.3\%\) and iFormer-S \(83.4\%\); Heracles-C-Base at \(6.5\) G achieves \(85.2\%\) versus Wave-ViT-B \(84.8\%\) and MaxViT-S \(84.5\%\); Heracles-C-Large at \(13.4\) G achieves \(85.9\%\) versus VOLO-D3 \(85.4\%\) and Wave-ViT-L \(85.5\%\); and Heracles-C-Huge at \(39.3\) G achieves \(86.4\%\) versus LiT-22B \(85.9\%\). The paper states that Heracles-C-small achieves state-of-the-art performance on ImageNet with \(84.5\%\) top-1 accuracy, and that Heracles-C-Large and Heracles-C-Huge further improve accuracy to \(85.9\%\) and \(86.4\%\), respectively [2403.18063].

The same work reports transfer-learning results on CIFAR-10, CIFAR-100, Flowers-102, and Cars-196: \(99.2\%\), \(91.1\%\), \(98.9\%\), and \(93.5\%\), respectively, comparing favorably to DeiT-B. In MS-COCO instance segmentation with Mask R-CNN \(1\times\) schedule on val2017, Heracles-C-S reaches \(AP^b=45.9\), \(AP^b_{50}=67.8\), \(AP^b_{75}=50.2\), \(AP^m=41.6\), \(AP^m_{50}=65.0\), and \(AP^m_{75}=45.2\). In time-series forecasting on ETTm1/2, ETTh1/2, Electricity, and Weather, using MSE and MAE at horizons \(T=96,192,336,720\), it achieves best or second-best on almost all tables; examples given are ETTm1 at \(T=96\), MSE \(0.326\) versus best \(0.338\), and Electricity at \(T=96\), MSE \(0.145\) versus best \(0.159\) [2403.18063].

Ablation results isolate the architectural contribution of the parallel design. Global SSM only gives \(84.4\%\), local SSM only gives \(84.0\%\), a series Hartley\(\rightarrow\)Conv arrangement gives \(84.16\%\), and the parallel Hartley\(\parallel\)Conv arrangement gives \(84.46\%\), the best result in Table A1. The paper attributes this to joint global/local streams injecting both translation equivariance and long-range inductive biases absent in pure attention. It also reports that early spectral blocks reduce token dimension before quadratic attention, and gives an A100 latency of \(14.5\) ms for Heracles-C-S versus \(14.3\) ms for GFNet-H-S at \(224\times224\) [2403.18063].

## 6. Heracles as state-conditioned diffusion middleware for humanoid control

In humanoid robotics, Heracles is a hierarchical controller built from three modules: high-level reference motions, a state-conditioned diffusion middleware, and a low-level physics tracker. The reference signal is a time-indexed trajectory \(\mathbf{m}_t\). The middleware is a lower-frequency \(25\) Hz planner \(f_\theta\) that observes the robot’s physical state \(\mathbf{p}_t\) and the commanded reference \(\mathbf{m}_t\), and synthesizes a short-horizon keyframe trajectory \(\boldsymbol{\tau}_t\in\mathbb{R}^{K\times D}\). The low-level policy \(\pi\) runs at \(50\) Hz on a densified version \(\mathbf{m}'_t\) of \(\boldsymbol{\tau}_t\) and the proprioceptive state, issuing joint-position commands executed via PD control at \(200\) Hz. The receding-horizon loop is: observe \(\mathbf{p}_t,\mathbf{m}_t\); generate \(\boldsymbol{\tau}_t=f_\theta(\mathbf{p}_t,\mathbf{m}_t)\); densify \(\boldsymbol{\tau}_t\rightarrow\mathbf{m}'_t\); execute \(\mathbf{m}'_t\) for the next \(N_{\mathrm{exec}}\) steps [2603.27756].

The middleware uses a continuous flow-matching diffusion model in residual space. With static baseline
\[
\beta_{t,k}=\mathbf{p}_t\qquad (k=0,\dots,K-1),
\]
the residual is
\[
\mathbf{r}_t=\boldsymbol{\tau}_t-\boldsymbol{\beta}_t
\qquad\Longrightarrow\qquad
\boldsymbol{\tau}_t=\boldsymbol{\beta}_t+\mathbf{r}_t.
\]
If \(\mathbf{p}_t\approx \mathbf{m}_t\), then \(\mathbf{r}_t\approx \mathbf{0}\), preserving identity-like behavior. Flow matching is defined with \(\mathbf{x}_0\) as normalized ground-truth residual, \(\mathbf{x}_1\sim\mathcal{N}(0,I)\), interpolation path
\[
\mathbf{x}_t=(1-t)\mathbf{x}_0+t\mathbf{x}_1,\qquad t\in[0,1],
\]
conditioning vector
\[
\mathbf{c}_t=[\mathbf{p}_t,\mathbf{m}_t],
\]
and velocity-matching loss
\[
\mathcal{L}_{\rm vel}
=
\mathbb{E}_{t,\mathbf{x}_0,\mathbf{x}_1}
\left\|
\hat{\mathbf{v}}(\mathbf{x}_t,t;\mathbf{c}_t)-(\mathbf{x}_1-\mathbf{x}_0)
\right\|_2^2.
\]
Inference begins from
\[
\mathbf{x}_{t_{\rm start}}
=
(1-t_{\rm start})\,\mathrm{normalize}(\mathbf{r}^{\rm init})
+
t_{\rm start}\,\boldsymbol{\varepsilon},
\qquad
\boldsymbol{\varepsilon}\sim\mathcal{N}(0,I),
\]
with directional warm-start residual
\[
r^{\rm init}_k=\frac{k}{K-1}\,(\mathbf{m}_t-\mathbf{p}_t),
\]
and integrates
\[
\frac{d\mathbf{x}}{dt}=\hat{\mathbf{v}}(\mathbf{x},t;\mathbf{c}_t)
\]
using a small number of Euler steps, for example 5 steps from \(t=0.9\to0\) [2603.27756].

The small-deviation regime is treated explicitly as an identity-map regime. Because the model is parameterized around the current state, and because the first token is “pinned” via inpainting,
\[
\mathbf{x}_t[0]=(1-t_{\rm next})\,\mathbf{r}_0+t_{\rm next}\,\boldsymbol{\epsilon},
\]
the learned field satisfies \(\hat{\mathbf{v}}(\mathbf{x},t;\mathbf{c})\approx0\) for \(\mathbf{x}\approx0\), causing
\[
f_\theta(\mathbf{p}_t,\mathbf{m}_t)\approx [\,\mathbf{p}_t,\ldots,\mathbf{p}_t].
\]
For large deviations, the same system transitions into generative synthesis: as \(\|\mathbf{p}_t-\mathbf{m}_t\|\) grows, the residual departs from zero, the directional warm start seeds a coarse straight-line plan, and the learned field edits it into a human-like recovery in very few steps, akin to SD-Edit. Because the pipeline runs in closed loop at \(25\) Hz, it can continuously replan and blend back toward the original reference once stability is regained [2603.27756].

Quantitatively, on 101 unseen motion sequences spanning locomotion, dance, martial arts, daily activities, acrobatics, and discontinuous clips, the reported Completion Rate is \(90.6\%\) for Heracles versus \(87.2\%\) for the best non-generative tracker and \(84.8\%\) for an MLP baseline; root orientation error is \(0.273\) rad, a \(-24.5\%\) change versus tracker alone, and root height error is \(0.076\) m, a \(-20\%\) change. On the fall-and-recovery subset, Completion Rate is \(90.0\%\) versus \(44.0\%\) for the MLP and \(69.8\%\) for VQ-VAE; joint-position error is \(1.41\) rad versus \(2.17\) rad for the MLP, and orientation error is \(0.243\) rad versus \(1.016\) rad. With a fixed iFSQ tracker, removing directional warm start reduces Completion Rate by \(3.8\%\), removing noisy-state augmentation reduces it by \(13.2\%\), and removing kinematics-aware loss weights reduces it by \(9.4\%\). Qualitatively, under severe pushes and out-of-distribution martial-arts sequences, Heracles is reported to remain anthropomorphic, exhibiting compensatory stepping, arm countermotions, and torso realignment, while pure trackers collapse or execute jerky corrections; on Unitree G1, it executes omnidirectional lie-to-stand recoveries from supine, lateral, and prone configurations without task-specific engineering [2603.27756].

Source: https://www.emergentmind.com/topics/heracles