Probabilistic Sampling of Balanced K-Means Clustering using Adiabatic Quantum Computing

Core Concepts

The proposed approach uses an adiabatic quantum computer to sample solutions of a balanced k-means clustering problem. By using an energy-based formulation, likely solutions are drawn from a Boltzmann distribution, and the calibrated posterior probability of each solution is estimated.

Abstract

The paper presents a quantum computing formulation of balanced k-means clustering that predicts well-calibrated confidence values and provides a set of alternative clustering solutions. The key highlights are: The clustering problem is formulated as an energy-based model, where the energy function corresponds to the k-means objective and the cluster size constraints. This allows embedding the problem into an adiabatic quantum computer (AQC) that samples solutions from the corresponding Boltzmann distribution. To address the challenges of temperature mismatch and hardware imperfections in the AQC, a reparametrization approach is used to compute posterior probabilities from the samples, avoiding the need for exact tuning of the sampling temperature. Extensive experiments on synthetic and real data (IRIS dataset) demonstrate the calibration of the approach using simulation as well as the D-Wave Advantage 2 AQC prototype. The results show that the proposed method can provide a set of high-probability clustering solutions along with their confidence estimates. The probabilistic clustering solutions and confidence scores can be used to identify ambiguous data points and provide alternative solutions, which can be beneficial for various computer vision applications like multi-object tracking, feature matching, and 3D reconstruction.

Stats

The energy function for the k-means objective is given as: Ek(X|Z) = (1/sk) * sum_i sum_j Zki * Zkj * (xi - xj)^T * (xi - xj) The quadratic penalty term for the cluster size constraints is: E(Z) = λ * ||Gz - d||^2

Quotes

"By formulating the k-means objective as a quadratic energy function, we embed the clustering task into the quantum-physical system of an AQC." "We recalibrate the samples of the clustering problem to address temperature mismatch, estimating the posterior probability for each solution, which allows us to identify ambiguous points and provides alternative solutions."

Key Insights Distilled From

Probabilistic Sampling of Balanced K-Means using Adiabatic Quantum Computing

by Jan-Nico Zae... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2310.12153.pdf

Deeper Inquiries

How can the proposed probabilistic clustering approach be extended to handle non-Gaussian data distributions or incorporate additional constraints beyond balanced cluster sizes

The proposed probabilistic clustering approach can be extended to handle non-Gaussian data distributions by incorporating different likelihood models that better represent the data distribution. One approach is to use a mixture of Gaussian distributions with different covariances for each cluster, allowing for more flexibility in capturing the data's underlying structure. Additionally, non-parametric models like kernel density estimation can be used to model the data distribution without assuming a specific parametric form. To incorporate additional constraints beyond balanced cluster sizes, the energy function used in the quantum clustering formulation can be modified to include these constraints. For example, constraints on cluster separability, cluster shape, or cluster density can be added to the energy function as penalty terms. By penalizing solutions that violate these constraints, the clustering algorithm can be guided towards solutions that satisfy the additional constraints while optimizing the clustering objective.

What are the potential limitations and challenges in scaling the quantum-based clustering approach to very large-scale problems, and how can these be addressed

Scaling the quantum-based clustering approach to very large-scale problems poses several potential limitations and challenges. One major challenge is the limited qubit connectivity and coherence times in current quantum hardware, which restrict the size and complexity of the clustering problems that can be solved efficiently. As the problem size increases, the number of qubits and interactions required also increases, leading to exponential growth in computational resources. To address these challenges, advancements in quantum hardware, such as increasing qubit connectivity and coherence times, are essential. Developing more efficient quantum algorithms tailored for clustering problems and optimizing the mapping of clustering problems to quantum hardware can also help improve scalability. Hybrid approaches that combine classical and quantum computing resources can be employed to handle larger problem sizes by offloading computationally intensive tasks to quantum processors.

Given the probabilistic nature of the quantum sampling, how can the uncertainty information be leveraged in downstream computer vision tasks like multi-object tracking, 3D reconstruction, or feature matching to improve their robustness and performance

The uncertainty information obtained from the probabilistic quantum sampling can be leveraged in downstream computer vision tasks to improve their robustness and performance. In multi-object tracking, the probabilistic clustering solutions can provide alternative hypotheses for object trajectories, allowing for more robust tracking in complex scenarios with occlusions or object interactions. By considering multiple likely solutions, the tracking algorithm can adapt to uncertainties and make more informed decisions. In 3D reconstruction, the uncertainty information from quantum clustering can help in identifying ambiguous correspondences between image features and 3D points. By considering the uncertainty in feature matching, the reconstruction algorithm can better handle noisy or ambiguous matches, leading to more accurate 3D reconstructions. Additionally, in feature matching tasks, the probabilistic clustering solutions can be used to generate diverse sets of candidate matches, improving the robustness of the matching process and reducing the risk of false positives.