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Warith Harchaoui

I am Warith Harchaoui

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Scientific Communications

Clustering Boosting

7 February 2020

Clustering is one of the most fundamental tasks in machine learning. This task consists in grouping similar objects together without any supervision, meaning there is no labels to guide this grouping. The goal is to find structure in data to facilitate further analysis. It is a desirable goal in data analysis, visualization and is often a preliminary step in many algorithms.

The subject of this blog post is to propose an answer to:
How can we combine several clustering algorithms outputs into one?

Indeed, as suggested by Andreas Weingessel et al. in An Ensemble Method for Clustering work, combining supervised classification outputs is well-studied and called boosting popularized and acclaimed in face detection with AdaBoost by Paul Viola and Michael Jones in practice and there is a thorough analysis in the theoretical book The Elements of Statistical Learning by Trevor Hastie et al.

Back to unsupervised classification (a.k.a. clustering), there is a considerable wealth of research and we would not dare to hope being unbiased nor exhaustive:

While being agnostic to the clustering algorithms, if we want to combine their output, we must choose a common format to represent them between:

Suppose you have a dataset of $N$ points to group into $K$ clusters and you are given $M$ soft clustering outputs, we want to find a way to eventually produce only one clustering aggregating all the other $M$ clustering outputs. To that end, we choose to re-format all clustering outputs into soft clustering thanks to one-hot-encoding (i.e. degenerated soft histograms for hard clustering) and also thanks to zero-padding in case of varying number of clusters. So we only manipulate soft clustering outputs without loss of generality. Moreover, unlike in supervised classification, for unsupervised classification (a.k.a clustering), the classes are anonymous : for hard clustering, the 11122 sequence over data is exactly the same clustering as 22211. For soft clustering, this means that randomly permuting the columns of the table does only change the clustering representation but not the clustering itself and this column-permutation invariance is called the switching label problem. This is a problem because it gives several representations for the same machine learning clustering object.

In terms of optimization, one can propose to cast this problem into: $$\min_{p,\sigma} \frac{1}{NM} \sum_{m=1}^M \sum_{i=1}^N \Vert p_i - \sigma_m(C_{m,i}) \Vert$$ where:

If we take a closer look to the proposed optimization problem, then we can see that the distance is not specified. Empirically, it turns out that the Hellinger distance which is specifically designed for histograms (positive $1$-sum vectors) is a good choice: $$ \forall a \in \mathbb{R}^K, b \in \mathbb{R}^K $$ $$\Vert a - b \Vert_\text{Hellinger} = \sum_{k=1}^K (\sqrt{a_k} - \sqrt{b_k})^2 = \Vert \sqrt{a} - \sqrt{b} \Vert_2^2 = 2 \times (1 - \sqrt{a}^\top \sqrt{b})$$ Mathematically, the Hellinger distance makes the histogram space (i.e. the positive $L_1$ sphere simplex) look spherical (in a the $L_2$ traditional euclidean sense) thanks to the square root which is cool for even rudimentary tool like a hyperplane that has more interpretable separation power cutting an $L_2$ sphere than cutting an $L_1$ sphere simplex as shown in this figure below.

Hyperplane cutting L1 or L2 spheres

Our optimization problem is very similar to the one of K-Means: (i) if $\sigma$ is given, then the optimal $p^*$ is found by a mean of square root probabilities and (ii) if the $p$ is given, then the optimal $\sigma^*$ is given by a cheap brute-force strategy to make the switching label problem disappear. Likewise, an alternated optimization scheme is efficient to (not optimally) solve our clustering aggregation (or boosting in reference to its supervised counterpart). Of course, there is no way to guarantee the optimal solution exactly like with K-Means but the solution quality cannot worsen w.r.t that objective until convergence in the same spirit as K-Means. Finally, we can also nicely adapt the K-Means++ strategy for careful initialization. Indeed, it seems we do K-Means over many simplistic or sophisticated clustering algorithms which is amusingly paying an ancient algorithm the respect that it deserves.

Favorite Academic Books

12 December 2019

In random order:

Small Data

6 April 2019

In 2012, I was privileged enough to attend to Divide-and-Conquer and Statistical Inference in Big Data given in French by Michael I. Jordan who is a prominent figure in Machine Learning. That visionary talk begun by establishing what can still be considered at the crux of Data Science today, especially back in a time when neither Hadoop nor Spark were popularized:

Consider $N$ the number of elements of the database (cardinality) and $D$ the dimensionality of each element.

  • Big Data regime $\frac{N}{D} \gg 1$: Developper's nightmare especially because data may not fit in memory but if the computations are doable, then statistics background give reasons for the systems to work very well especially since the rise of stochastic optimization methods;

  • Small Data regime $\frac{N}{D} \ll 1$ or $\frac{N}{D} \simeq 1$: Statistician's nightmare especially because of the mathematical curse of dimensionality but the computations are easier than previously.

NB: This has not been said that way by Michael I. Jordan but only simplified by myself

Since then, data storage problems have been nicely solved at least thrice thanks to the decrease of hardware's costs (and thus the increase of available computational power especially with the rise of GPUs), increase of data access speed and software solutions like HDFS and Spark (for both distributed file and in-memory computer systems) on the developper's side but the curse of dimensionality remains on the statistician's side. Computations have also been improved but not sufficiently energy-wise (that is beyond that current blog post scope).

Enforcing data structure seems to be an efficient way to cope with the curse of dimensionality issue. For example, if one considers the huge progress accomplished with images from the 1990s to the 2010s, then one sees that methods with hierarchical convolutions have been the key to unlock good classification results at the historical ImageNet competition ($\sim 10^6$ images database with $\sim 10^6$ pixels each thus $\frac{N}{D} \simeq 1$ which corresponds to a Small Data regime). Likewise, words ordering turns out to be the key to Natural Language Processing representations but despite worldwide Research attention in Genetics, Machine Learning tends to be uneffective yet: $1$ human being $D \simeq 10^{12}$ nucleotides and databases are limited to the worldwide population ($\sim 10^9$), thus $\frac{N}{D} \ll 1$ which corresponds to a Small Data regime once again. We did not yet leverage any desirable but unknown structure among those nucleotides to get an easier Big Data regime.

The take-home-message of this post is:

Indeed, structure translates data knowledge and human expertise into an understandable statistical language that is more suitable for a computer. By the way, if $N$ is not large ($\sim 10^ 1$ or $\sim 10 ^ 2$) then statistical estimators do not have much relevance in such very Small Data regime ($N \leq 10 ^ 2$).

What is a Data Scientist?

10 February 2017

One saying goes: A data scientist is someone who is better at Statistics than any software engineer and better at Software Engineering than any statistician.

For beginners with some Math and Programming background, here are two books to get the right mix of skills between Statistics and Computer Sciences:

Useful micro-courses for basic tools in a few hours: kaggle.com/learn

Programming skills in C#, R, Java (including Clojure) and even C/C++ for experts are bonuses.

Skills in Statistics and more importantly some French bon sens  (common sense, discernment) seem mandatory before trying new fancy algorithms on real-world problems.

Non-negative Weights Linear Models for Classification

22 December 2016

We suppose that data are normalized (by mean and variance or between 0 and 1). For certain applications, such as diagonal Mahalanobis Metric Learning, it is useful to have a linear-model-implementation code for classification that has non-negative weights. Without sign constraints on the weights, there exists several nice out-of-the-shelves implementations freely accessible in the web, for example:

but none of them authorize a built-in mechanism to impose sign constraints on the weights.

One solution could be to modify these implementations and do a Projected Gradient Descent in the primal by zeroing the negative weights at each step. But this process is inconvenient because the nice convergence properties are then lost.

In this post, I would like to suggest another way to benefit from existing linear models implementations without modifying their codes. In a previous post (Regularized Bias in (Stochastic) Gradient Descent) I showed how to handle the regularized bias by adding a supplementary constant feature equal to $1000$. To impose non-negative weights, we gain augment the data set by all the $B$-hot vectors possible with $B \gg 1$ (e.g. $B=1000$) but without bias. A $B$-hot vector is a vector with zeros everywhere except for one bin which contains $B$. If we look at the SVM formulation, we have for weights $\mathbf{w}$, labels $y_i \in \{ -1,1 \}$ and examples $\mathbf{x}_i$

$$ \min_{\mathbf{w}} \frac{1}{2} \Vert \mathbf{w} \Vert^2_2$$ such that for each $i$ indexing the dataset $$y_i \mathbf{w}^\top \mathbf{x}_i \ge 1 - \xi_i $$

If $\mathbf{x}_i$ is a $B$-hot vector with a positive label ($y_i= 1$), then we get: $$\mathbf{w}_k \ge \frac{1- \xi_i}{B} \simeq 0$$ for each $k$ indexing the dimensions of $\mathbf{w}$. So, up to the slack variable $\xi_i$ which is most of the time zero or less than one, we have constrained the sign of $\mathbf{w}_k$. In practice, we can zero out the negative values of the libraries' output.

This slight modification of regular SVMs also works for logistic regression (because the hinge and logistic losses have almost the same behavior) and is easy for sparse representation as $B$-hot vectors are ridiculously sparse.

Detection is not Classification

22 December 2015

There is a fondamental difference between Classification and Detection, let's say for the binary case. On the one hand, Classification distinguishes two well-defined classes and everything is symmetric. On the other hand, Detection aims to detect one class versus the rest of the world, so the problem is not symmetric. For example, getting a nice dataset for Classification is fairly easy, it is enough to sample points from the positive and the negative classes. But in the Detection case, it is very difficult to sample "enough" points from the negative cases because representing the rest of the world is fairly difficult. For example, in the latent-SVM paper from Felzenszwalb et al for Object Detection there is a tedious Hard Negative Mining step to cope with this problem.

More recently, the Convolutional Neural Networks revolution started by the seminal work of Alex Krizhevsky's AlexNet, handles this problem by leveraging a huge number of classes (thousands) in a simulataneous One-vs-Rest fashion. As we speak, there is no explicit discriminative framework for Detection without Classification. Maybe the One-class SVM framework could do but I did not experiment with it.

Regularized Bias in (Stochastic) Gradient Descent

21 December 2015

It is useful to remind the reader that data should always be normalized, à la StandardScaler of scikit-learn (which substract the mean and divide by the standard deviation for each dimension of your data). In this context, the optimization papers almost never precise how to handle the Bias. In the excellent explanatory C++ code of Léon Bottou, we have several modes to regularize or not the Bias.

What does it mean exactly? Data Science practionners have often heard that the Bias is a detail and that it is enough to add a "1" dimension to the data. But in fact the Bias plays a crucial role during the optimization. For example, for Linear Support Vector Machines or Logistic Regression, the Bias sets the sign of the prediction driving the direction of the gradient step.

One way to handle the Bias is: add a "1000"-dimension instead of a "1"-dimension. This way, the true Bias found at the end of the optimization should be divided by 1000. In fact, the Bias will be regularized but just a little which is a trade-off between an ease of optimization (no special case for the Bias) and having no regularization for the Bias (which is instable).

Miscellaneous