Mathematical Sciences: Wavelets and their Applications to Neural Network Theory, Vision, and Image Processing

[9410859] Kon研究员和他的同事研究小波及其在神经网络理论和视觉图像处理中的应用。最近的研究表明，小波作为神经网络的激活函数是有用的，它比目前常用的s型激活函数更快地适应所需的任务。小波方法的成功已经在使用径向基函数的重要优化问题中得到了令人印象深刻的证明。该项目的工作为构建高效学习的神经网络提供了更好的理论基础，并研究了模拟“智能”任务（例如视觉图像识别）的神经网络的最小复杂性。研究人员还研究了基于小波的图像压缩算法的构建。他们研究了信号压缩的优化，旨在从函数和它们的小波变换之间的映射的角度改进对这种信号压缩如何工作的理论理解。这里提出的数学应用与使用神经网络技术创建“智能”系统密切相关。这项技术在很多方面模拟了生物神经系统的运作。目前的工作是基于在某些情况下生物神经元的行为在人工神经系统中被模拟得太好。Girosi和Poggio等人的研究表明，具有“局部激活函数”的网络可以更好地实现神经网络中的预期行为，即输出最终会随着非常大的输入而减少的网络。与使用神经网络影响适当的输入输出行为相关的另一个问题是了解某些期望任务的基本复杂性，以及神经网络如何最好地实现这种复杂性水平。这样的任务（例如，物体的视觉识别）已经可以由生物神经网络执行，并且具有可以以相对精确的方式定义和研究的复杂性，以确定人工神经网络需要多复杂才能模拟生物网络的行为。此外，使用小波的信号压缩工作具有双重影响。第一个涉及压缩技术的改进，例如，将大量的视频、音频或其他数据存储在压缩媒体（如压缩光盘、硬盘或随机存取存储器）上。第二个原因是，如果这些数据被压缩到更小的尺寸，就可以更容易地从中提取某些类型的信息。例如，如果心脏学数据被压缩成一种更容易操纵的方式，那么使用计算机就更容易找到不规则心跳（或心律失常的前兆）的证据。这些和其他应用使得这里研究的数据压缩问题在数学信号处理中具有很高的优先级。

9410859 Kon The investigator and his colleagues study wavelets and their applications to neural network theory and processing of visual images. It has been shown recently that wavelets are useful as activation functions in neural networks, permitting faster adaptation to required tasks than do sigmoidal activation functions, in common use up to now. The success of wavelet methods has already been impressively demonstrated in important optimization problems using radial basis functions. The work of this project develops better theoretical underpinnings for the construction of neural nets that learn efficiently, and studies minimal complexities of neural nets that emulate "intelligent" tasks, e.g., recognition of visual images. The investigators also study the construction of wavelet-based algorithms for image compression. They examine the optimization of signal compression, aiming at an improved theoretical understanding of how such signal compression works from the standpoint of the mapping between functions and their wavelet transforms. Applications of the mathematics proposed here are closely related to the creation of "intelligent" systems using neural network technology. This technology in many ways emulates the operation of biological nervous systems. The present work is based on indications that the actions of biological neurons have in some cases been too well emulated in artificial neural systems. It has been shown in the work of Girosi and Poggio and others that attaining desired behavior in neural networks can be better achieved with networks that have "localized activation functions," i.e., ones for which output eventually decreases with very large inputs. An additional issue related to effecting proper input-output behavior using neural networks is knowledge of the basic complexity of certain desired tasks, and how neural networks can best achieve this level of complexity. Such tasks (e.g., the visual recognition of object s) can already be performed by biological neural networks, and have a complexity that can be defined and studied in a relatively precise way, to determine how complicated artificial neural networks need to be in order to emulate the behaviors of the biological networks. In addition, the work on signal compression using wavelets has a two-fold impact. The first involves the improvement of techniques for compression, e.g., storage of large quantities of video, audio or other data on compact media (such as compact discs, hard disks, or random access memory). The second is related to the fact that if such data are compressed to smaller sizes, certain kinds of information may be extracted from them more easily. For example, it is easier to find evidence of an irregular heartbeat (or precursors to arhythmia) using a computer if the cardiological data have been compressed in a way that allows them to be more easily manipulated. These and other applications have made the data compression issues that are studied here an area with high priority in mathematical signal processing.