Spheres Dataset: Multi-Domain Benchmark

• The Spheres Dataset is a collection of influential benchmarks across machine learning, music information retrieval, geoscience AI, and astronomy.
• It provides controlled environments with detailed methodologies, from geometric adversarial analysis to isolated orchestral recordings and multimodal Earth system evaluations.
• These benchmarks enhance research by offering structured datasets that facilitate performance evaluation in adversarial learning, source separation, multi-step reasoning, and astrophysical imaging.

The term “Spheres Dataset” spans several distinct and influential datasets and benchmarks across scientific domains, each structuring complex data modalities and tasks according to the geometry, physical layout, or semantic domains of “spheres.” Major contributions include the synthetic Adversarial Spheres for geometric analysis of adversarial robustness in machine learning, The Spheres Dataset for multitrack orchestral music research, Omniearth-Bench’s Spheres benchmark for Earth systems AI, and the SPHERE database for exoplanet imaging. The following sections delineate structure, methodology, and impact for each major instantiation.

1. Synthetic Adversarial Spheres Dataset

The synthetic Spheres dataset, introduced by Gilmer et al. in "Adversarial Spheres" (Gilmer et al., 2018), is a paradigmatic example of a minimal, analytically tractable manifold that exposes the high-dimensional geometric roots of adversarial vulnerability in classification.

• Definition: The task is binary classification between two concentric $d$-dimensional spheres. With radii $r_1 < r_2$, two surfaces are defined as $S_1 = \{x \in \mathbb{R}^d : \|x\|_2 = r_1\}$ (inner) and $S_2 = \{x \in \mathbb{R}^d : \|x\|_2 = r_2\}$ (outer).
• Data Generation: Each datapoint $x$ is sampled by drawing $z \sim N(0, I_n)$, then normalizing and scaling: $x = r_1 \cdot z/\|z\|_2$ (inner) or $x = r_2 \cdot z/\|z\|_2$ (outer), selected with probability $1/2$ each. The binary label $y$ encodes sphere membership ($y=0$ for inner, $y=1$ for outer).
• Parameters: Experiments set $r_1 = 1.0$, $r_2 = 1.3$, and $d = n = 500$. Both infinite (“online”) and finite-sample settings (sizes $N \in \{10^3,\ldots, 10^6\}$) were explored.
• Preprocessing: No transformation or normalization beyond enforced $\ell_2$-norm. Input points to models are exactly as sampled from $S_1$ or $S_2$.

2. Geometric Theorem: Classification Error vs Adversarial Robustness

A central contribution of the Adversarial Spheres work is a tight, model-independent upper bound on typical adversarial perturbation size in the presence of nonzero test error, grounded in high-dimensional isoperimetry (Gilmer et al., 2018).

• Notation: Let $E \subset S_1$ denote misclassified points on the inner sphere, $\mu(E)$ their probability mass (error rate), and $d(E)$ the expected $\ell_2$ distance from a random $x \sim S_1$ to the nearest element of $E$.
• Theorem: If a classifier achieves accuracy $p$ on $S_1$ ($\mu(E) = 1-p$), then

$d(E) \leq \frac{\Phi^{-1}(p)}{\sqrt{d}},$

where $\Phi$ is the standard normal CDF. For fixed error $\epsilon$, $d(E) = O(1/\sqrt{d})$. Hence, for any constant nonzero error rate, most points reside within $O(1/\sqrt{d})$ of a misclassification.

• Implication: In high dimension, adversarial perturbations of vanishing norm ($O(1/\sqrt{d})$) are inevitable unless the test error decays super-exponentially in $d$. This establishes that adversarial example proximity is a geometric and probabilistic certainty, not an artifact of specific classifier architectures.

3. The Spheres Dataset for Orchestral Source Separation

"The Spheres Dataset: Multitrack Orchestral Recordings for Music Source Separation and Information Retrieval" (Garcia-Martinez et al., 26 Nov 2025) presents the first publicly released multitrack orchestral dataset optimized for machine learning in MIR (Music Information Retrieval), especially supervised source separation.

• Content: Over one hour of orchestral recordings (Tchaikovsky’s Romeo and Juliet, Mozart’s Symphony No. 40), full chromatic scales and solo excerpts, captured by 23 microphones (ambient, main, close spot, section/spot).
• Recording Protocol: Each instrument track is recorded in isolation using standardized reference tracks and a live conductor for tight temporal alignment. This yields precisely aligned stems with known, controllable bleed (crosstalk), and detailed room impulse responses (RIRs) for each seat-microphone pairing, facilitating studies of acoustic propagation and dereverberation.
• Data Structure: Directory structure mirrors musical pieces and microphone roles, with each subfolder containing per-instrument WAV files (48 kHz/24 bit). RIR measurements are provided as .npy arrays and visualized PDFs.
• Splits and Benchmarks: Primary recommendation is Tchaikovsky for training, Mozart for evaluation due to limited corpus size; baseline experiments use X-UMX (Sawata et al. 2021) for orchestral family separation and debleeding, evaluated via SDR, SIR, SAR, and ISR metrics.
• Applications: Enables research in supervised/semi-supervised separation, polyphony detection, score-informed separation, deep localization (via RIRs), spatial audio rendering, and dereverberation. The controlled bleed paradigm supports the development and validation of separation architectures exploiting known crosstalk matrices and spatial arrangements.

4. OmniEarth-Bench Spheres Dataset for Geoscience AI

OmniEarth-Bench’s Spheres Dataset (Wang et al., 29 May 2025) is the first comprehensive multimodal evaluation suite for geosystem-aware AI, systematically covering Earth’s six spheres—Atmosphere, Lithosphere, Oceansphere, Cryosphere, Biosphere, Human-Activities—and their cross-sphere interactions.

• Scope: 29,779 annotated samples across 33 modalities (EO images, in-situ waveforms, derived biophysical indices), partitioned by scientific domain (sphere), scenario, capability (perception, reasoning, chain-of-thought), and subtask.
• Task Design: Encompasses VQA (27,082 MCQ, 2,697 visual grounding), and is built as an evaluation suite (not for training/validation splitting). Balances classes and image resolutions per subtask.
• Annotation Workflow: 2–5 PhD-level domain experts per sphere define tasks and gold labels, with crowd annotators for scale, then iterative expert–crowd adjudication (final expert accuracy 90–97% per sphere).
• Metrics: Per-sphere and cross-sphere accuracy, visual grounding metrics (Precision/Recall@IoU), and F1_CoT for multi-step reasoning chains.
• Findings: State-of-the-art MLLMs (OpenAI GPT-4o, Gemini-2.0, InternVL3) do not exceed 35% aggregate accuracy; cross-sphere tasks yield marked performance decline (GPT-4o: 0.04% cross-sphere accuracy), revealing fundamental limitations in current model architectures for integrated geoscience reasoning.
• Use Cases: Benchmarks geoscientific AI for environmental monitoring, disaster prediction, and climate analysis; supports research in multi-modal, cross-domain model alignment at planetary scale.

5. Spheres and “SPHERE”: Astronomical Imaging Datasets

While not itself called the “Spheres Dataset,” the “spherical” database (Samland, 9 Sep 2025) for the ESO VLT/SPHERE instrument represents a major public resource for direct imaging of exoplanets and circumstellar disks.

• Coverage: $~6000$ IRDIS dual-band imaging, $1000$ IRDIS polarimetric, $4500$ IFS data cubes, plus planned addition of ZIMPOL, IRDIS-LSS, and SAM modes.
• Data Architecture: PostgreSQL/SQLite schema with cross-matched Gaia stellar parameters, observing conditions, calibration records. Automated pipelines apply wavelength, astrometric, and photometric calibration; TRAP post-processing delivers posterior detection maps and $5\sigma$ contrast curves.
• Query Interoperability: Designed for end-to-end integration with VIP, pyKLIP, and IRDAP for PSF subtraction and polarization analysis. Enables efficient survey population studies and detailed companion characterization at uniform sensitivity.

6. Domain-Specific Contributions and Prospective Directions

Each instantiation of a Spheres Dataset operationalizes the “sphere” concept in a distinct domain for unique research leverage:

• Adversarial Spheres: Precise geometric characterization of the adversarial phenomenon in high dimensions, highlighting data manifold structure as a key driver of classifier vulnerability (Gilmer et al., 2018).
• Orchestral Spheres: Real multitrack data supporting rigorous, source-aware benchmarking of separation/dereverberation under controlled acoustic conditions and known bleeding topology (Garcia-Martinez et al., 26 Nov 2025).
• Geoscience Spheres: Holistic, multimodal benchmarking for AI spanning basic perception to chain-of-thought reasoning across interconnected earth system components; sets new baselines for model evaluation (Wang et al., 29 May 2025).
• Astronomical SPHERE: Systematic, large-scale structured astrophysics data with seamless reduction/analysis workflows, propelling population analyses and heterogeneous science cases (Samland, 9 Sep 2025).

A plausible implication is that the “Spheres Dataset” concept provides a recurring organizational approach that enables benchmark construction and experimental control in otherwise heterogeneous settings, by exploiting either physical geometry (as in adversarial spheres), physical layout (as in audio/microphone spheres), or semantic subdivision (as in geoscience domains or astronomical observation modes).

7. Comparative Overview

Dataset Name	Primary Domain	Structure/Modality	Core Tasks	Key Reference
Adversarial Spheres	ML Theory	2 concentric spheres in $\mathbb{R}^d$	Binary classification, adversarial bounds	(Gilmer et al., 2018)
Spheres Orchestral	Music/MIR	23-chan. multitrack audio	Source separation, debleeding, RIR analysis	(Garcia-Martinez et al., 26 Nov 2025)
OmniEarth Spheres	Geoscience AI	33 geospatial data types	Multimodal VQA, reasoning, visual grounding	(Wang et al., 29 May 2025)
spherical (SPHERE) DB	Astronomy	Imaging cubes, catalogs	Exoplanet detection, spectral extraction	(Samland, 9 Sep 2025)

Each Spheres Dataset serves as a domain benchmark, facilitating new methods, controlled diagnostics, and deep exploration of underlying structures in machine learning, physical modeling, and environmental or musical information processing.