How is Compressed Sensing going to change Machine Learning ?

Compressed Sensing (CS) is a new framework developed by Emmanuel Candes, Terry Tao and David Donoho. To summarize, if you acquire a signal in some basis that is incoherent with the basis in which you know the signal to be sparse in, it is very likely you will be able to reconstruct the signal from these incoherent projections.

Terry Tao, the recent Fields medalist, does a very nice job at explaining the framework here. He goes further in the theory description in this post where he mentions the central issue of the Uniform Uncertainty Principle. It so happens that random projections are on average incoherent, within the UUP meaning, with most known basis (sines, polynomials, splines, wavelets, curvelets …) and are therefore an ideal basis for Compressed Sensing. [ For more in-depth information on the subject, the Rice group has done a very good job at providing a central library of papers relevant to the growing subject: http://www.dsp.ece.rice.edu/cs/ ]

The Machine Learning community has looked at Random Projections before, for instance:

  • Experiments with Random Projections for Machine Learning by Fradkin and Madigan (KDD-03.)
  • Face Recognition Experiments with Random Projection by Goel, Bebis and Nefian
  • Dimensionality reduction by random mapping: Fast similarity computation for clustering by S. Kaski (Proceedings of IEEE International Joint Conference on Neural Networks, 1998.)
  • but while they seem to give somewhat comparable results with regards to PCA, the number of contributions on the subject does not seem overwhelming. Maybe one of the reason is that in most papers cited above, the main theoretical reason for using Random projections lies with the Johnson-Lindenstrauss (JL) lemma. As a consequence, most random matrices used in these publications come from the Database world and not from the newer framework of Compressed Sensing (a list of these matrices and their properties can be found in the middle of this page). The uncanny reliance on Random projections within the JL lemma and in the Compressed Sensing setting was explained by Richard Baraniuk, Mark Davenport, Ronald DeVore, and Michael Wakin in this paper entitled: A simple proof of the restricted isometry property for random matrices. However, the most interesting fallout from this comparison between JL and CS comes in the form of the contribution by Richard Baraniuk and Michael Wakin in Random projections of smooth manifolds. I’ll let the abstract speak for itself:

    We propose a new approach for nonadaptive dimensionality reduction of manifold-modeled data, demonstrating that a small number of random linear projections can preserve key information about a manifold-modeled signal……As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on M are well-preserved under the mapping \phi. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in RN. Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and curvature of the manifold. In addition to recovering faithful approximations to manifold-modeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.

    It looks as though, as a result, Universal Dimensionality Reduction is achieved by some properly chosen random projections. In the case of data living in a low dimensional manifold, the JL lemma states that the number of random projections is proportional to the number of points or samples from that manifold, on the other hand, CS seems to show that the number of random projections is proportional to the characteristic of the manifold only.

    The results highlighted by Wakin and Baraniuk are very compelling but there is another appealing reason to Random Projections: Robustness. While trying to mimick nature in the learning process, one cannot but be amazed at the reliability of the biological system. On the other hand, even the researchers that model these processes do not realize or point out that this robustness is in part due to random projections. Case in point, the excellent work of Thomas Serre, Aude Oliva and Tomaso Poggio culminating in a paper describing a biology inspired model of brain that shows its ability to process information in a feedforward fashion. The modeling is new in that, in this area of science, there is a central issue as to whether the brain works in a one-way process or with many loops. In the paper, the feature dimension reduction model (which is what this process is) uses random projections as I pointed out recently.

    Because of the intrinsic dimension reduction capability, Mike Wakin has also shown efficient nearest neighbor searches using few random projections (see figure 3 of this paper). I could go on but the point is that since CS is a revolution in the world of signal/feature acquisition and processing (see the analog-to-information A2I site ) one cannot but wonder aloud how this will affect Machine Learning in general.

    Introspectionism as a Disease

    In the AI-related parts of machine learning, it is often tempting to examine how you do things in order to imagine how a machine should do things. This is introspection, and it can easily go awry. I will call introspection gone awry introspectionism.

    Introspectionism is almost unique to AI (and the AI-related parts of machine learning) and it can lead to huge wasted effort in research. It’s easiest to show how introspectionism arises by an example.

    Suppose we want to solve the problem of navigating a robot from point A to point B given a camera. Then, the following research action plan might seem natural when you examine your own capabilities:

    1. Build an edge detector for still images.
    2. Build an object recognition system given the edge detector.
    3. Build a system to predict distance and orientation to objects given the object recognition system.
    4. Build a system to plan a path through the scene you construct from {object identification, distance, orientation} predictions.
    5. As you execute the above, constantly repeat the above steps.

    Introspectionism begins when you believe this must be the way that it is done.

    Introspectionism arguments are really argument by lack of imagination. It is like saying “This is the only way I can imagine doing things, so it must be the way they should be done.” This is a common weak argument style that can be very difficult to detect. It is particularly difficult to detect here because it is easy to confuse capability with reuse. Humans, via experimental tests, can be shown capable of executing each step above, but this does not imply they reuse these computations in the next step.

    There are reasonable evolution-based reasons to believe that brains minimize the amount of computation required to accomplish goals. Computation costs energy, and since a human brain might consume 20% of the energy budget, we can be fairly sure that the evolutionary impetus to minimize computation is significant. This suggests telling a different energy-conservative story.

    An energy consevative version of the above example might look similar, but with very loose approximations.

    1. The brain might (by default) use a pathetically weak edge detector that is lazily refined into something more effective using time-sequenced images (since edges in moving scenes tend to stand out more).
    2. The puny edge detector might be used to fill a rough “obstacle-or-not” fill map that coarsens dramatically with distance.
    3. Given this, a decision about which direction to go next (rather than a full path) might be made.

    This strategy avoids the need to build a good edge detector for still scenes, avoids the need to recognize objects, avoids the need to place them with high precision in a scene, and avoids the need to make a full path plan. All of these avoidances might result in more tractable computation or learning problems. Note that we can’t (and shouldn’t) say that the energy conservative path “must” be right because that would also be introspectionism. However, it does exhibit an alternative showing the failure of imagination in introspectionism on the first approach.

    It is reasonable to take introspection derived ideas as suggestions for how to go about building a (learning) system. But if the suggestions don’t work, it’s entirely reasonable to try something else.