meaning of linear representations

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Linear representation

A homomorphism $\pi$ of a group (respectively an algebra, ring, semi-group) $X$ into the group of all invertible linear operators on a vector space $E$ (respectively, into the algebra, ring, multiplicative semi-group of all linear operators on $E$). If $E$ is a topological vector space, then a linear representation of $X$ on $E$ is a representation whose image contains only continuous linear operators on $E$. The space $E$ is called the representation space of $\pi$ and the operators $\pi(x)$, $x\in X$, are called the operators of the representation $\pi$.

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Title: the linear representation hypothesis and the geometry of large language models.

Abstract: Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

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nLab linear representation

Linear algebra.

homotopy theory , (∞,1)-category theory , homotopy type theory

flavors: stable , equivariant , rational , p-adic , proper , geometric , cohesive , directed …

models: topological , simplicial , localic , …

see also algebraic topology

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geometry of physics – homotopy types

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homotopy coherent category theory

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(∞,1)-category

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infinitesimal interval object

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fundamental group

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fundamental ∞-groupoid

fundamental groupoid

• path groupoid

fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos

fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos

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nerve theorem

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homotopy hypothesis -theorem

Characters of linear representations

Characteristic classes of linear representations, related concepts.

A linear representation is a representation on a category of vector spaces or similar ( Vect , Mod , etc.)

This is the most common flavor of representations . One sometimes considers representations on objects other than linear spaces (such as permutation representations ) but often these are called not representations but actions .

See characters of linear representations .

Under the Atiyah-Segal completion map linear representations of a group G G induce topological K-theory classes on the classifying space B G B G . Their Chern classes are hence invariants of the linear representations themselves.

See at characteristic class of a linear representation for more.

permutation representation

category of representations

• Tammo tom Dieck , Representation theory , 2009 ( pdf )

For more see the references at representation theory .

Last revised on February 1, 2019 at 14:18:03. See the history of this page for a list of all contributions to it.

Linear Representation: Contemporary Use

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• First Online: 18 March 2020
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• Tommaso Empler 15  

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1140))

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• International and Interdisciplinary Conference on Image and Imagination

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“Linear representation” has been used, in the past of western history and near east, with celebrative functions and a mode of representation where a perception of depth is used linked to the overlapping of elements present in the scene and a consequent visualization in side elevation. A rediscovery of “linear representation” occurs in contemporary age, where it assumes a technical use in temporary and permanent environmental solutions or in new media, defining a new research field for representation applied to architecture, graphics and design, better known as “environmental graphics”. In this context, a particular development takes place on some ephemeral structures such as yard fences, with horizontal development, or in facades of buildings.

• Linear representation
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• Perceptive depth cues
• Gestalt principles

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Costantin, A.L.: Compendio di geometria pratica applicata al disegno lineare. Officina tipografica di Giovanni Enrici e comp., Saluzzo (1843)

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De Fiore, G.: Dizionario del Disegno. La Scuola Editore, Brescia (1967)

Empler, T.: Grafica e comunicazione ambientale. Dei – Tipografia del genio Civile, Roma (2012)

Gagnazzo, S.:. Muro di Berlino, tutta la sua street art (2017). https://www.lastampa.it/2017/07/28/cultura/muro-di-berlino-tutta-la-sua-street-art-9EG0CAfluiHUUHoDlVY6EP/pagina.html . Accessed 12 Feb 2019

Norberg-Schulz, C., Norberg-Schulz, A.M.: Genius Loci. Paesaggio, ambiente, architettura. Electa - Documenti di architettura, Milano (1992)

Schreiber, G.: Il disegno lineare corso pratico per artisti e industriali e specialmente per le scuole tecniche normali e professionali. Loescher, Torino (1872)

Valeri, V.: Corso di Disegno 2. La Nuova Italia Editrice, Firenze (1990)

Vallortigara, G., Panciera, N.: Cervelli che contano. Adelphi Edizioni, Milano (2014)

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Department of History, Representation and Restoration of Architecture, Sapienza University of Rome, Piazza Borghese 9, 00186, Rome, Italy

Tommaso Empler

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Correspondence to Tommaso Empler .

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University of Sassari, Alghero, Italy

Enrico Cicalò

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Empler, T. (2020). Linear Representation: Contemporary Use. In: Cicalò, E. (eds) Proceedings of the 2nd International and Interdisciplinary Conference on Image and Imagination. IMG 2019. Advances in Intelligent Systems and Computing, vol 1140. Springer, Cham. https://doi.org/10.1007/978-3-030-41018-6_40

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LISR: Learning Linear 3D Implicit Surface Representation Using Compactly Supported Radial Basis Functions

• Atharva Pandey Indian Institute of Technology, Jodhpur
• Vishal Yadav Indian Institute of Technology, Jodhpur
• Rajendra Nagar Indian Institute of Technology, Jodhpur
• Santanu Chaudhury Indian Institute of Technology, Jodhpur

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3.2.1: Introduction to Linear Relationships

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Let's explore some relationships between two variables.

Exercise \(\PageIndex{1}\): Number Talk: Fraction Division

Find the value of \(2\frac{5}{8}\div\frac{1}{2}\).

Exercise \(\PageIndex{2}\): Stacking Cups

We have two stacks of Styrofoam cups.

• One stack has 6 cups, and its height is 15 cm.
• The other stack has 12 cups, and its height is 23 cm.

How many cups are needed for a stack with a height of 50 cm?

Exercise \(\PageIndex{3}\): Connecting Slope to Rate of Change

1. If you didn’t create your own graph of the situation before, do so now.

2. What are some ways you can tell that the number of cups is not proportional to the height of the stack?

3. What is the slope of the line in your graph? What does the slope mean in this situation?

4. At what point does your line intersect the vertical axis? What do the coordinates of this point tell you about the cups?

5. How much height does each cup after the first add to the stack?

Andre starts babysitting and charges $10 for traveling to and from the job, and $15 per hour. For every additional hour he works he charges another $15. If we graph Andre's earnings based on how long he works, we have a line that starts at $10 on the vertical axis and then increases by $15 each hour. A linear relationship is any relationship between two quantities where one quantity has a constant rate of change with respect to the other.

We can figure out the rate of change using the graph. Because the rate of change is constant, we can take any two points on the graph and divide the amount of vertical change by the amount of horizontal change. For example, take the points \((2,40)\) and \((6,100)\). They mean that Andre earns $40 for working 2 hours and $100 for working 6 hours. The rate of change is \(\frac{100-40}{6-2}=15\) dollars per hour. Andre's earnings go up $15 for each hour of babysitting.

Notice that this is the same way we calculate the slope of the line. That's why the graph is a line, and why we call this a linear relationship. The rate of change of a linear relationship is the same as the slope of its graph.

With proportional relationships we are used to graphs that contain the point \((0,0)\). But proportional relationships are just one type of linear relationship. In the following lessons, we will continue to explore the other type of linear relationship where the quantities are not both 0 at the same time.

Glossary Entries

Definition: Linear Relationship

A linear relationship between two quantities means they are related like this: When one quantity changes by a certain amount, the other quantity always changes by a set amount. In a linear relationship, one quantity has a constant rate of change with respect to the other.

The relationship is called linear because its graph is a line.

The graph shows a relationship between number of days and number of pages read.

When the number of days increases by 2, the number of pages read always increases by 60. The rate of change is constant, 30 pages per day, so the relationship is linear.

Exercise \(\PageIndex{4}\)

A restaurant offers delivery for their pizzas. The total cost is a delivery fee added to the price of the pizzas. One customer pays $25 to have 2 pizzas delivered. Another customer pays $58 for 5 pizzas. How many pizzas are delivered to a customer who pays $80?

Exercise \(\PageIndex{5}\)

To paint a house, a painting company charges a flat rate of $500 for supplies, plus $50 for each hour of labor.

• How much would the painting company charge to paint a house that needs 20 hours of labor? A house that needs 50 hours?
• Draw a line representing the relationship between \(x\), the number of hours it takes the painting company to finish the house, and \(y\), the total cost of painting the house. Label the two points from the earlier question on your graph.

3. Find the slope of the line. What is the meaning of the slope in this context?

Exercise \(\PageIndex{6}\)

Tyler and Elena are on the cross country team.

Tyler's distances and times for a training run are shown on the graph.

Elena’s distances and times for a training run are given by the equation \(y=8.5x\), where \(x\) represents distance in miles and \(y\) represents time in minutes.

• Who ran farther in 10 minutes? How much farther? Explain how you know.
• Calculate each runner's pace in minutes per mile.
• Who ran faster during the training run? Explain or show your reasoning.

(From Unit 3.1.4)

Exercise \(\PageIndex{7}\)

Write an equation for the line that passes through \((2,5)\) and \((6,7)\).

(From Unit 2.3.3)