Are there any image classification algorithms which are not neural networks? Image classification is the task of assigning one of $n$ previously known labels to a given image. For example, you know that you will be given a couple of photos and each single image has exactly one of $\{cat, dog, car, stone\}$ in it. The algorithm should say what the photo shows.

The benchmark dataset for image classification is ImageNet; especiall thy large scale visual recognition challenge (LSVRC). It has exactly 1000 classes and a huge amount of training data (I think there is a down-sampled version with about 250px x 250px images, but many images seem to be from Flicker).

This challenge is typically solved with CNNs (or other neural networks).

Is there any paper which tries an approach which does not use neural networks in LSVRC?

To clarify the question: Of course, there are other classification algorithms like $k$ nearest neighbors or SVMs. However, I doubt they work at all for that many classes / that much data. At least for $k$-NNs I'm sure that prediction would be extremely slow; for SVMs I guess both fitting and prediction would be much to slow (?).
Part of the problem with answering this question is there are actually two questions.  The first:

Are there any image classification algorithms which are not neural networks?

Yes, lots. But now the actually question:

Is there any paper which tries an approach which does not use neural networks in LSVRC?

In your question, you rule out methods such as kNN and SVM because of speed. Bag of Words is one method used to solve this problem.  MATLAB has a good demonstration (http://www.mathworks.com/help/vision/examples/image-category-classification-using-bag-of-features.html).  But BoW incorporates k-means clustering, so that may not fit your needs.

There are some other interesting image classification methods such as texture analysis. TA is being researched as a way to classify malevolence of disease in medical images (such as tumors).  Here is a commonly referenced paper:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2701316/

Here's an overview of image classification:
http://www.tandfonline.com/doi/full/10.1080/01431160600746456