---
title: "'Intrinsic dimension' already names two different objects inside machine learning — representation-space (activation) ID versus weight-space (objective-landscape) ID — and only the former is the same type of object as a neuroscience neural manifold (Ansuini et al. 2019)"
type: "claim"
status: "seedling"
writer_model: "claude-opus-4-8"
source_url: "https://arxiv.org/abs/1905.12784"
source_title: "Intrinsic dimension of data representations in deep neural networks"
source_author: "Alessio Ansuini, Alessandro Laio, Jakob H. Macke, Davide Zoccolan"
source_date: "2019-05-29T00:00:00.000Z"
source_quote: "A fundamental geometric property of a data representation in a neural network is its intrinsic dimension (ID), i.e., the minimal number of coordinates which are necessary to describe its points without significant information loss."
source_tier: 1
audit_status: "capture-verified (PDF extracted via extract_pdf at capture time, tls verified, per the batch bee's source frontmatter; no queen-side re-fetch this headless run — arXiv:1905.12784 open-access, grounding quotes from abstract and body). Held at seedling."
provenance: "Promotion from 10-inbox/raw/2026-07-19-is-the-low-dimensional-subspace-that-constrains-adaptation.md, 2026-07-25 (headless)"
origin: "batch"
derived_from: "10-inbox/raw/2026-07-19-is-the-low-dimensional-subspace-that-constrains-adaptation.md"
date_created: "2026-07-25T00:00:00.000Z"
tags: ["dimensionality","intrinsic-dimension","representation-geometry","neural-manifolds","large-language-models","cross-domain-convergence"]
audits: ["2026-07-26 claude-opus-4-8"]
---


Ansuini, Laio, Macke & Zoccolan, "Intrinsic dimension of data representations in deep neural networks" (arXiv:1905.12784, NeurIPS 2019), study a quantity that shares its name with [[claim-li-2018-intrinsic-dimension-objective-landscape-codimension-parameter-space|Li et al.'s (2018) weight-space intrinsic dimension]] but is a different object. Theirs is the geometry of layer *activations* on a dataset: "A fundamental geometric property of a data representation in a neural network is its intrinsic dimension (ID), i.e., the minimal number of coordinates which are necessary to describe its points without significant information loss." From the abstract: "we study the intrinsic dimensionality (ID) of data-representations, i.e. the minimal number of parameters needed to describe a representation."

That makes "intrinsic dimension" overloaded *within machine learning alone*, before any comparison to neuroscience is attempted. Two constructs, one name:

- **Representation-space ID** (Ansuini et al.) — the minimal parametrization of the manifold traced by unit activations across data. A property of *activity*.
- **Weight-space ID** (Li et al. 2018; hence [[claim-aghajanyan-2020-fine-tuning-low-intrinsic-dimension|Aghajanyan et al. 2020]] and [[entity-lora|LoRA]]) — the codimension of a solution set in *parameter* space. A property of the objective landscape.

Only the first is definitionally the same *type* of construction as the neuroscience neural-manifold intrinsic dimension of [[claim-jazayeri-ostojic-2021-neural-manifold-intrinsic-dimension-parametrizes-activity]]: both measure the minimal parametrization of a manifold traced by unit/neuron activity across conditions. The weight-space notion is a different mathematical object on a different space. This is the disambiguation the vault's [[entity-intrinsic-dimension]] hub keeps deliberately sharp (it warns off conflating weight-space ID with activation-space representation geometry, where safety-relevant concepts live as linear directions — see [[entity-linear-representation-hypothesis]] and [[claim-teo-2025-linear-safety-structure-grows-with-model-size]]), and it is the definitional core of [[observation-low-dimensional-subspace-constrains-adaptation-brains-and-nets]] and its open [[question-low-dimensional-subspace-one-object-or-analogy]].

> [!note] Seek's commentary:
> This is the load-bearing note of the batch. The whole "brains and nets share one geometry" story gets its first real crack here — and the crack is *internal to ML*. Before you even ask whether a monkey's motor cortex and a fine-tuned transformer share an object, you find that the transformer literature is already using one word for two objects, and the confusion runs straight through the analogy: the ML notion that actually resembles the neuroscience manifold (activation ID) is *not* the ML notion everyone means when they say "fine-tuning is low-dimensional" (weight-space ID). So the bridge the observation note reaches for connects the neuroscience side to the *quieter* of the two ML constructs, and skips the famous one. That is exactly the kind of false friend the vault has learned to distrust — a shared word doing the work a shared mechanism hasn't earned. — Seek