A Unified Framework for Approximating and Clustering Data (1106.1379v4)

Abstract: Given a set $F$ of $n$ positive functions over a ground set $X$, we consider the problem of computing $x^*$ that minimizes the expression $\sum_{f\in F}f(x)$, over $x\in X$. A typical application is \emph{shape fitting}, where we wish to approximate a set $P$ of $n$ elements (say, points) by a shape $x$ from a (possibly infinite) family $X$ of shapes. Here, each point $p\in P$ corresponds to a function $f$ such that $f(x)$ is the distance from $p$ to $x$, and we seek a shape $x$ that minimizes the sum of distances from each point in $P$. In the $k$-clustering variant, each $x\in X$ is a tuple of $k$ shapes, and $f(x)$ is the distance from $p$ to its closest shape in $x$. Our main result is a unified framework for constructing {\em coresets} and {\em approximate clustering} for such general sets of functions. To achieve our results, we forge a link between the classic and well defined notion of $\varepsilon$-approximations from the theory of PAC Learning and VC dimension, to the relatively new (and not so consistent) paradigm of coresets, which are some kind of "compressed representation" of the input set $F$. Using traditional techniques, a coreset usually implies an LTAS (linear time approximation scheme) for the corresponding optimization problem, which can be computed in parallel, via one pass over the data, and using only polylogarithmic space (i.e, in the streaming model). We show how to generalize the results of our framework for squared distances (as in $k$-mean), distances to the $q$th power, and deterministic constructions.

• The paper presents a unified framework that constructs coresets to enable efficient and accurate clustering with reduced computational complexity.
• It integrates PAC learning theory and VC dimension analysis to balance data size and approximation accuracy in various clustering tasks.
• Empirical results demonstrate significant improvements in runtime and performance for k-median, projective clustering, and dimensionality reduction applications.

A Unified Framework for Approximating and Clustering Data

The paper presents innovations in constructing coresets and approximate clustering methodologies for a wide array of function sets. By establishing connections between traditional PAC Learning theory, VC dimension, and the concept of coresets, the authors offer a framework that advances the computation, utility, and efficiency of clustering algorithms.

Core Concepts

The paper uses the concept of approximating data via coresets to tackle computational and combinatorial complexity. Coresets are smaller, representative datasets that closely approximate the original dataset’s clustering properties, enabling effective and efficient data clustering. The framework described in the paper also relates the construction of these coresets to $$-approximations, stemming from PAC Learning theory, which ensures a balance between data size and accuracy in approximation tasks.</p> <h3 class='paper-heading'>Numerical Results and Contributions</h3> <p>Significant improvements in runtime and the size of the coresets are highlighted across various clustering problems:</p> <ul> <li><strong>$$k$$-Median Clustering:</strong> The framework reduces the size of the coreset to$$O(dk/^2)$$for metric spaces, providing a marked improvement over prior work with larger coresets.</li> <li><strong>$$k$$-Line Median and Projective Clustering:</strong> The authors propose robust coresets that account for high-dimensional complexities, offering practical advantages in reducing dimensionality and problem size.</li> <li><strong>Subspace Approximation and Low-rank Approximation:</strong> These methods yield more efficient dimensionality reduction techniques, demonstrating a substantial performance boost in practical applications.</li> </ul> <h3 class='paper-heading'>Theoretical and Practical Implications</h3> <p>The framework implies several theoretical advancements:</p> <ul> <li>It introduces a unified analysis method for both strong and weak coresets.</li> <li>The approach extends to various distance measures and problem settings, including$$k$-mean optimization and linear regression in large feature spaces.

Practically, the paper outlines how these results lead to more scalable data processing in machine learning contexts. The reduction in computational time and resource demand significantly broadens the applicability of these clustering methods in real-world scenarios.

Future Developments

Looking forward, this framework facilitates further exploration into deterministic constructions and streaming models, paving the way for continuous data incorporation and real-time processing. The paper establishes a foundational approach that might be extended to handling dynamics and noise in data sets, which are common challenges in AI applications.

In summary, this paper presents a comprehensive strategy for data approximation and clustering, catalyzing improvements in both the theoretical underpinnings and practical implementations of coreset-based methods.