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Interpretable Neural Marked Statistics for Cosmological Inference

This paper proposes an interpretable neural marked statistics framework that uses contrastive learning to generalize non-linear field reweighting, significantly improving cosmological parameter constraints and breaking degeneracies compared to classical methods.

Original authors: Federico Semenzato, Benjamin D. Wandelt, Michele Liguori, Alvise Raccanelli

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Federico Semenzato, Benjamin D. Wandelt, Michele Liguori, Alvise Raccanelli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, three-dimensional sponge made of invisible matter. Astronomers want to understand how this sponge was formed and what it's made of (like dark energy or the mass of neutrinos). To do this, they usually look at how clumps of matter are spaced apart from each other.

Traditionally, scientists have used a tool called the "power spectrum." Think of this like listening to a song and only analyzing the volume of each note (low, mid, high). It's a great way to understand the basic rhythm of the universe, but it misses the complex harmonies and the unique texture of the sound. In the universe, these "missing harmonies" are the subtle, non-linear patterns that hold the most secrets about how the cosmos evolved.

The Problem: Missing the "Texture"

The standard method (the power spectrum) is like looking at a map of the universe and only counting how many dots are in a circle. It works well for simple, smooth distributions. But as the universe gets older, matter clumps together in complex shapes—filaments, voids, and knots. The standard method starts to lose its ability to read these complex shapes accurately.

The Solution: "Marking" the Map

The authors propose a new way to look at this map. Instead of just counting dots, they want to "mark" different parts of the map with special labels based on their local shape.

Imagine you are looking at a forest.

The Old Way: You count how many trees are in a 1-mile square.
The New Way (Marked Statistics): You look at a specific tree and ask, "Is this tree in a dense thicket? Is it standing alone in a clearing? Is it part of a long, winding row?" You then give that tree a "weight" or a "mark" based on its surroundings.
The Result: When you count the trees again, the ones in the dense thicket count for more, or maybe the ones in the clearing count differently. This "re-weighting" folds the complex 3D shape information back into a simple 2D count, revealing hidden details.

The Innovation: Teaching a Computer to Find the Best Marks

In the past, scientists had to guess which "marks" were best. They would say, "Let's give extra weight to dense areas" or "Let's highlight empty spaces." This was like trying to tune a radio by guessing the right station.

This paper introduces a neural network (a type of AI) that learns the perfect way to "mark" the universe.

The Architecture: The AI looks at the universe through different "lenses" (mathematical filters) that detect specific shapes:
- Isotropic (Round): How dense is the area overall?
- Dipole (Directional): Is there a flow or a gradient?
- Quadrupole (Elongated): Is the matter stretched out like a cigar or flattened like a pancake?
The Learning Process: The AI tries millions of different ways to combine these shapes into a "mark." It uses a technique called contrastive learning.
- The Analogy: Imagine a teacher showing the AI a picture of a forest and asking, "What is the temperature?" The AI tries to guess. If it guesses wrong, the teacher says, "No, that picture belongs to a cold forest, not a hot one." The AI learns to adjust its "marking" so that the picture of the forest perfectly matches the "temperature" (the cosmological parameters).
The Residual Trick: To make sure the AI isn't just repeating what we already know, the researchers force it to only learn new information. They subtract the "old" knowledge (the standard power spectrum) from the AI's answer. The AI is only rewarded if it finds something the old method missed.

The Results: Sharper Focus

When they tested this on simulated universes (digital models of the cosmos), the results were impressive:

Breaking the Tie: In cosmology, two major parameters (how much matter there is, $\Omega_m$ , and how clumpy the universe is, $\sigma_8$ ) often get confused with each other. It's like trying to tell if a blurry photo is dark or out of focus. The new method "rotates" the data, separating these two factors so they can be measured independently.
Better Precision: The method improved the measurement of the "clumpiness" ( $\sigma_8$ ) by nearly 3 times and the amount of matter ( $\Omega_m$ ) by nearly 2 times compared to the best previous methods.
Interpretability: Unlike many AI models that are "black boxes" (we know they work but not why), this one is transparent. The researchers can look at the AI's "brain" and see exactly which shapes it decided were important. They found that the AI mostly focused on the overall density (the "round" lens) but used the directional and elongated lenses to fine-tune the details at the boundaries of cosmic structures.

Summary

The paper presents a new, AI-driven tool that "tags" the cosmic web with intelligent labels based on local shapes. By teaching the AI to find the best tags without human bias, they can extract much more information from the universe's structure than ever before, specifically solving a long-standing problem where two key cosmic measurements were stuck together. It's a step toward a clearer, more detailed picture of our universe's history.

Technical Summary: Interpretable Equivariant Marks for Contrastive Cosmological Inference

Problem Statement
Recovering cosmological information beyond the power spectrum is a central goal for upcoming Large-Scale Structure (LSS) surveys (e.g., SPHEREx, DESI, Euclid). While the power spectrum is optimal for Gaussian fields, the late-time matter density exhibits non-Gaussian signals that two-point statistics alone cannot capture. Traditional approaches to extract this information face significant hurdles:

Higher-order correlators: Climbing the ladder of $n$ -point correlators is computationally expensive in configuration space, and perturbation theory breaks down in the non-linear regime.
Field-level inference: Simulation-based inference and field-level neural summaries bypass explicit statistics but often sacrifice interpretability and require high-fidelity simulations.
Classical marked statistics: Existing methods reweight the density field with a spatial mark $M(x)$ to fold higher-order correlations into the two-point level. However, these marks are typically hand-designed with fixed functional forms (e.g., smoothed-density responses) or optimized only at a single fiducial cosmology, limiting their adaptability and interpretability across the parameter space.

Methodology
The authors propose a framework that replaces hand-designed marks with a learned, cosmology-agnostic, and interpretable neural marking scheme. The framework consists of two primary components:

Interpretable Learned Mark Architecture:
- The mark $M(x)$ is constructed as a translation-equivariant scalar response derived from the matter density field $\delta(x)$ .
- Spherical Harmonic Filtering: The density field is decomposed into channels using spherical-harmonic-aware filters ( $\ell \in \{0, 1, 2\}$ $ℓ \in {0, 1, 2}$ ) in Fourier space. These filters produce rotation-invariant scalar channels:
  - $\ell=0$ : A signed monopole response ( $E_0$ ).
  - $\ell=1$ : A vector-like response with invariant amplitude ( $E_1$ ).
  - $\ell=2$ : Two scalar invariants derived from a traceless symmetric tensor ( $E_2$ for anisotropy strength, $I_3$ for distinguishing prolate/oblate shapes).
- Processing: These invariant channels are processed by independent Multi-Layer Perceptrons (MLPs) and combined additively with a cross-interaction term. The output is passed through a softplus activation to ensure positivity.
- Initialization: The network is initialized such that $M(x) \approx 1$ , meaning training begins from the unmarked field. This allows the learned mark to be decomposed into distinct morphological contributions.
Contrastive Learning Objective:
- The framework employs a contrastive learning objective (InfoNCE) to align marked summaries with cosmological parameters $\theta$ in a shared latent space.
- Summary Vectors: The summary includes the power spectrum of the unmarked field ( $P_{\delta\delta}$ ) and the marked field ( $P_{\Delta\Delta}$ ), along with the cross-spectrum ( $P_{\delta\Delta}$ ) and correlation coefficient.
- Residualization: To ensure the mark learns complementary information rather than simply reproducing the power spectrum, the marked summary embedding ( $z_M$ ) is residualized against the unmarked embedding ( $z_\delta$ ) to produce an orthogonal component $z_\perp$ .
- Loss Function: The loss aligns $z_\perp$ with the parameter embedding $z_\theta$ using a learned Mahalanobis metric, while separating it from negative candidates sampled from the prior (real, global synthetic, and local synthetic negatives).
- Training Strategy: The unmarked branch is pre-trained and frozen; the marker network and parameter embedder are then fine-tuned using the contrastive loss.

Key Contributions

Generalization of Marked Statistics: The paper introduces a neural architecture that generalizes the concept of marked statistics, moving beyond fixed functional forms to a learned transformation that remains inspectable in configuration space.
Interpretability via Equivariance: By enforcing local SO(3)-equivariance and using spherical harmonic filters, the model extracts features with clear physical meanings (monopole, dipole, quadrupole responses) that can be directly visualized and analyzed.
Contrastive Alignment: The use of a contrastive objective with residualization ensures the learned mark specifically targets information orthogonal to the standard power spectrum, breaking degeneracies that traditional statistics cannot.

Results
The method was evaluated on the Quijote N-body simulation suite ( $z=0$ ) with 5 varying cosmological parameters ( $\Omega_m, \Omega_b, h, n_s, \sigma_8$ ).

Fisher Information Constraints: At $k_{\max} = 0.2 \, h\text{Mpc}^{-1}$ , the learned mark tightens marginalized constraints by 2.9 $\times$ for $\sigma_8$ and 1.8 $\times$ for $\Omega_m$ compared to classical marks. Crucially, it breaks the canonical $\Omega_m - \sigma_8$ degeneracy, rotating the joint posterior contour to allow simultaneous improvement on both parameters.
Generalization (MSE): Across a Latin-hypercube test set, the learned mark reduces the parameter Mean Squared Error (MSE) by 1.45 $\times$ over the best classical mark. It demonstrates robust generalization, particularly for $\sigma_8$ , though classical marks remain slightly more stable for $\Omega_m$ in specific $(p, \delta_s)$ configurations.
Latent Geometry: Principal Component Analysis (PCA) of the learned embedding reveals that the first two principal components align almost perfectly with the $\sigma_8$ and $\Omega_m$ directions, confirming that the contrastive objective successfully recovers the dominant axes of cosmological information.
Introspection: Analysis of the trained mark shows that the isotropic ( $\ell=0$ ) channel dominates the response, effectively recovering a non-linearly reweighted density, while anisotropic channels provide subdominant corrections at structure boundaries.

Significance
The paper claims that this approach opens the door to more powerful and interpretable summary statistics for cosmological inference. By replacing hand-crafted transformations with a learned, physically motivated architecture, the method bridges the gap between the high constraining power of field-level neural networks and the interpretability of traditional statistics. The ability to inspect the learned mark in configuration space allows researchers to understand which local morphological features (e.g., voids, filaments, anisotropic structures) carry the richest cosmological signal. The authors position this as a step toward flexible inference tools that can reduce simulation costs and improve the scientific return of galaxy surveys, with potential extensions to other parameters like primordial non-Gaussianity.