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#### Nonlinear Receptive Field Analysis: Making Kernel Methods Interpretable

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##### Citation

Kienzle, W., Macke, J., Wichmann, F., Schölkopf, B., & Franz, M. (2007). *Nonlinear Receptive Field Analysis: Making Kernel Methods Interpretable*. Poster presented at
Computational and Systems Neuroscience Meeting (COSYNE 2007), Salt Lake City, UT, USA.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-CECB-D

##### Abstract

Identification of stimulus-response functions is a central problem in systems neuroscience and related areas.
Prominent examples are the estimation of receptive fields and classification images [1]. In most cases, the
relationship between a high-dimensional input and the system output is modeled by a linear (first-order) or
quadratic (second-order) model. Models with third or higher order dependencies are seldom used, since
both parameter estimation and model interpretation can become very difficult.
Recently, Wu and Gallant [3] proposed the use of kernel methods, which have become a standard tool in
machine learning during the past decade [2]. Kernel methods can capture relationships of any order, while
solving the parameter estmation problem efficiently. In short, the stimuli are mapped into a high-dimensional
feature space, where a standard linear method, such as linear regression or Fisher discriminant, is applied.
The kernel function allows for doing this implicitly, with all computations carried out in stimulus space.
As a consequence, the resulting model is nonlinear, but many desirable properties of linear methods are
retained. For example, the estimation problem has no local minima, which is in contrast to other nonlinear
approaches, such as neural networks [4].
Unfortunately, although kernel methods excel at modeling complex functions, the question of how to interpret
the resulting models remains. In particular, it is not clear how receptive fields should be defined in
this context, or how they can be visualized. To remedy this, we propose the following definition: noting
that the model is linear in feature space, we define a nonlinear receptive field as a stimulus whose image in
feature space maximizes the dot-product with the learned model. This can be seen as a generalization of the
receptive field of a linear filter: if the feature map is the identity, the kernel method becomes linear, and our
receptive field definition coincides with that of a linear filter. If it is nonlinear, we numerically invert the
feature space mapping to recover the receptive field in stimulus space.
Experimental results show that receptive fields of simulated visual neurons, using natural stimuli, are correctly
identified. Moreover, we use this technique to compute nonlinear receptive fields of the human fixation
mechanism during free-viewing of natural images.