This post will be the last one in my mini-series “A Similarity Space for Shapes” about joint work with Margit Scheibel. So far, I have described the overall data set, the correlation between distances and dissimilarities, and the well-shapedness of conceptual regions. Today, I will finally take a look at interpretable directions in this similarity space.

# Author: Lucas Bechberger

## A Similarity Space for Shapes (Part 3)

Today, I would like to continue my little series about recent joint work with Margit Scheibel on a psychologically grounded similarity space for shapes. In my first post, I outlined the data set we worked with, and in my second post, we investigated how well the dissimilarity ratings are reflected by distances in the similarity spaces. Today, I’m going to use the categories from our data set to analyze whether conceptual regions are well-formed.

## What is a “Principle Component Analysis”?

I’m currently in the process of writing the background chapter on Machine Learning for my dissertation. In the context of doing that, I took a closer look at a widely used feature extraction technique called “Principle Component Analysis” (PCA). It can be described either on the intuitive level (“it finds orthogonal directions of greatest variance in the data”) or on the mathematical level (“it computes an eigenvalue decomposition of the data matrix”). What I found most difficult to understand was how these two descriptions are related to each other. This is essentially what I want to share in today’s blog post. Continue reading “What is a “Principle Component Analysis”?”

## A Similarity Space for Shapes (Part 2)

In one of my last blog posts, I have introduced a data set of shapes which I use to extract similarity spaces for the shape domain. As stated at the end of that blog post, I want to analyze these similarity spaces based on three predictions of the conceptual spaces framework: The representation of dissimilarities as distances, the presence of small, non-overlapping convex regions, and the presence of interpretable directions. Today, I will focus on the first of these predictions. More specifically, we will compute the correlation between distances in the MDS-spaces to the original dissimilarities and compare this to three baselines. This will help us to see how efficiently the similarity spaces represent shape similarity.

## Four Psychological Theories about Concepts

I’ve already introduced the notion of a concept in two older blog posts (see here and here) to set the stage for the conceptual spaces framework and to motivate why concepts are useful. In short, a concept is a mental representation of a category of things in the world. For example, the concept *apple* ties together all knowledge we have about apples in general, such as their typical shapes and sizes, as well as what they can be used for (e.g., eating or throwing). In order to embed the conceptual spaces framework a bit more in the overall area of concept research, I will today sketch four psychological theories about concepts (based on the great overview by Murphy [1]) and show how they can be related to the conceptual spaces framework.

Continue reading “Four Psychological Theories about Concepts”