Are there useful applications of SVD that use only the smallest singular values? In a number of singular value decomposition (SVD) applications, for example Latent Semantic Indexing, only the biggest singular values are used to make searches and calculate distances.
Are there useful applications that drop the biggest singular values and use only the smallest ones? 
 A: *

*Slow feature analysis (SFA) uses the smalles Eigenvalues of the covariance matrix of temporal differences to find the slowest features in a time series,

*Minor component analysis (MCA) uses the smallest components in a probabilistic setting--here, not directions of variations are found but constraints,

*Extreme component analysis (XCA) is a combination of probabilistic PCA and MCA,

*In Canonical Correlation Analysis (where you analyse the correlation btw two different data sets), the smaller components of the correlation matrix correspond to so called "private" spaces. These represent the subspaces of each variable which do not correlate linearly with each other.

A: It acts like a highpass filter in a slightly different space.  
There is lots of linear data, and in many cases you are looking for that linear relationship, so a low-pass (high-blocking) filter lets you retain the important part.  
For non-linear data, usually stuff that you have applied the simple methods without success to, the high-pass means that you throw away the unimportant (linear) part.
This makes me wonder about computational photography and edging.  Thanks..
A: Total Least Squares regression (aka Orthogonal Distance regression) uses the singular vector corresponding to the smallest singular value of the augmented predictor/criterion matrix.  
When there is only one dependent variable (i.e., when $k = 1$), both equation 12.3-5 in my Golub & Van Loan (first edition), and the final equation and Octave code in the "Algebraic point of view" section of the Standard account, use only the singular vector corresponding to the smallest singular value to get the vector of regression coefficients.
A: Yes, there are. I'm currently working with a professor on a research project where we try to predict very short-term stock market changes based on real-time tweets from Twitter. Unfortunately, the majority of what people say on Twitter about the companies we are tracking is useless rambling. In other words, the largest singular values are useless. 
Our plan is to use the biggest singular values to flag vast amounts of garbage tweets so that we can delete them. The remaining tweets and their text content (the small singular values) are candidates for a variable selection process. 
We are trying find the needles in a haystack, and dropping the largest singular values is like setting the hay on fire. 
A: It's a bit of a stretch, but consider the portfolio optimization problem: minimize $w^{\top} \Sigma w$ subject to $w^{\top} w \ge 1$. You can think of this as the minimum variance portfolio with an $\ell_2$ constraint. After applying Lagrange Multiplier method, you find that $w$ should be the eigenvector associated with the smallest eigenvalue of $\Sigma$. Since $\Sigma$ is typically the sample covariance $(1/N) \sum_{1\le i\le N} X_i X_i^{\top}$, where the $X_i$ have been centered, you can view this problem as an SVD computation where the singular vector associated with the smallest singular value is of importance. Like I said, it's a bit of a stretch.
A: There is an interesting LSA related paper which concludes that discarding (a lot of) first SVD features can improve results on semantic tests like TOEFL - "Extracting Semantic Represenations from Word Cooccurrence Statistics: Stop-lists, Stemming and SVD" (Bullinaria and Levy, 2012)
A: I'm not aware of any.  The smallest singular values correspond to modes that don't contribute much to the reconstruction of the original matrix, or to use the PCA interpretation, don't describe much of the variance in the data.  Typically,  the modes with smaller singular values are just noise.  This doesn't rule out the possibility that some meaning could be found in them, but I think it would be highly dependent on the data which make up the original matrix and — honestly — pretty unlikely. 
