[Paper Review] Simple Classification using Binary Data

Deanna Needell, Rayan Saab+1 more|arXiv (Cornell University)|Jul 6, 2017

Sparse and Compressive Sensing Techniques31 references18 citations

TL;DR

This paper proposes a low-complexity framework for classifying data using only binary (1-bit) measurements, leveraging random projections via a Gaussian matrix A to encode data into sign patterns. It demonstrates effective classification with theoretical guarantees on correct classification probability, particularly in a two-class, two-dimensional setting, offering a foundation for efficient, hardware-friendly learning systems.

ABSTRACT

Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this work, we study the problem of data classification from binary data and propose a framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, and provide a theoretical analysis for a simple case. We hope that our framework and analysis will serve as a foundation for studying similar types of approaches.

Motivation & Objective

• To develop a classification framework that operates solely on binary representations of data, reducing computational and hardware resource demands.
• To demonstrate that effective classification is possible using only sign information from random linear projections, even in highly quantized settings.
• To provide a mathematically tractable method that can be theoretically analyzed and extended to more complex learning problems.
• To establish theoretical bounds on classification accuracy under mild assumptions, particularly in low-dimensional, two-class scenarios.

Proposed method

• The method uses a random matrix A with i.i.d. standard Gaussian entries to project high-dimensional data into a lower-dimensional space.
• Binary measurements are obtained via the sign operator applied to the projected data, resulting in Q = sign(AX), which encodes the side of hyperplanes each data point lies on.
• A two-stage classification approach is proposed: training on labeled binary data to learn decision rules, followed by inference on new binary data.
• The classification rule is based on counting agreements between the sign patterns of new data and training data across multiple random hyperplanes.
• Theoretical analysis derives a lower bound on the probability of correct classification for two-dimensional, two-class problems under mild distributional assumptions.
• The framework is validated numerically on synthetic data and real-world datasets, including handwritten digit and facial recognition tasks.

Experimental results

Research questions

• RQ1 Can effective classification be achieved using only 1-bit measurements of data, without full signal recovery?
• RQ2 What is the theoretical guarantee on classification accuracy when using random projections and sign-based measurements?
• RQ3 How does the performance of the proposed method scale with the number of random projections and data dimensionality?
• RQ4 Can the framework be extended to multi-class problems and real-world applications like digit and face recognition?
• RQ5 What is the impact of the random projection matrix structure on classification reliability and robustness?

Key findings

• The proposed method achieves high classification accuracy on synthetic and real datasets, including MNIST and facial recognition benchmarks, using only binary data.
• In the two-dimensional, two-class case, the paper derives a non-trivial lower bound on the probability of correct classification, which depends on the data distribution and the number of random projections.
• Theoretical analysis shows that under mild assumptions, the classification error probability decays exponentially with the number of random projections.
• The framework is computationally efficient, requiring only sign operations and counting, making it suitable for low-power and embedded hardware applications.
• Numerical experiments confirm that the method performs well even when the number of projections is small, indicating robustness to low-dimensional measurements.
• The method outperforms baseline approaches in scenarios with extreme quantization, demonstrating the viability of 1-bit learning.