# DRY considered harmful: an evolutionary alternative

A growing number of programmers have observed that the object-oriented programming (OOP) dogma DRY (“don’t repeat yourself”) is sometimes actually harmful. A number of alternatives have been proposed, all of which involve thinking clearly and carefully. Think about the right level of abstraction at which shared implementation makes sense. Think about avoiding shared logic and ideas, rather than shared code. Think about bounded contexts, and avoid straying across them.

These are all great ideas, but they all require careful thought. Unfortunately, in many situations developers lack the ability (especially places that had massively duplicated code pre-DRY) or time (especially research settings) to think through these delicate questions.

Evolution offers a useful analogy to think through this problem in such settings. Genetic evolution enables adaptability through duplication. Quantitative traits are encoded across hundreds or thousands of parallel genetic variants. These polygenic traits are able to evolve much more rapidly than monogenic traits where a single genetic variant underlies all variation in a population. Sexual reproduction breaks up highly-correlated genetic variants (“linkage disequilibrium”) so that they can evolve independently to assist the fitness of a species. And the code for these traits is not only duplicated within a single individual’s genetic code: they are duplicated among all the individuals in a population.

Deep neural networks also evolve via a learning algorithm, such as stochastic gradient descent, and they utilize duplication to adapt their weights. In recent years, it has been discovered that wide “overparameterized networks”, with more activations than the number of samples, have better learning dynamics. This extra capacity is not strictly necessary to represent complex functions, but networks are more easily able to move their weights to such functions with this overcapacity.

In software development, when we tolerate duplicated code rather than force shared implementation, each copy can evolve in its natural direction, unencumbered by other use cases. Of course, this approach has a cost: code bloat. Fortunately, evolution offers us a solution: pruning. In biological evolution, natural selection filters out harmful genetic variants. And increasingly, pruning is used in deep learning to speed up inference of over-parameterized networks. Notably, pruning in both evolution and deep learning works very well with even simple strategies. It is often hard to find an accurate neural net, but once you have one, it’s easy to find hidden units that can be discarded.

This is a useful approach in R&D as well, both at an individual level and at a team level. When getting started on a new research project, it’s useful to copy-and-paste old code, recklessly changing it to meet one’s needs. As the project matures, one can readily identify duplicated code, and then start merging them. Similarly, it is hard for a development team to figure out and agree in advance on which code logic can rely on shared implementation. Yet, so long as all programmers have the discipline to “clean up after themselves” — to look out for old duplicated code and remove it — code quality can be maintained. Code deduplication (while it may not be fun) is actually intellectually easy, because it comes with the benefit of hindsight.

# The Newton Schulz iteration for matrix inversion

The Newton Schulz method is well-known, and the proof of convergence is widely available on the internet. Yet the derivation of the method itself is more obscure. Here it is:

We seek the zero of $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined as follows:
$\begin{array}{l} f(X) = X^{-1} - A. \end{array}$

The derivative of $f$ at $X$, applied to matrix $B$, operates on $B$ as follows:
$\begin{array}{l} f'(X)[B] = -X^{-1} B X^{-1}. \end{array}$

We can prove that $f'^{-1}$ at $X$, applied to matrix $B$, operates on $B$ as follows:
$\begin{array}{l} f'^{-1}(X)[B] = -X B X. \end{array}$

To see this, notice that
$\begin{array}{l} B \\ = f'^{-1}(X)\Big[f'(X)[B]\Big] \\ = -X \Big[-X^{-1} B X^{-1}\Big] X \\ = B. \end{array}$

The Newton method for root finding has at each iterate:
$\begin{array}{l} X_{t+1} \\ = X_t - f'^{-1}(X_t)\Big[f(X_t)\Big] \\ = X_t - f'^{-1}(X_t)\Big[X^{-1} - A\Big] \\ = X_t - X_t[X^{-1}_t-A] X_t \\ = X_t - [-X_t + X_t A X_t] \\ = 2 X_t - X_t A X_t \end{array}$

# Thresholding sparse matrices in Matlab

Here are the methods I tried:

function [tA] = hard_threshold(A, t)

tic;
tA = sparse(size(A));
tA(abs(A) >= t) = A(abs(A) >= t);
toc;

clear tA;
tic;
tA = A;
tA(abs(tA) < t) = 0;
toc;

clear tA;
tic;
tA = A;
find_A = find(A);
find_tA = find(abs(A) >= t);
victim_tA = setdiff(find_A, find_tA);
tA(victim_tA) = 0;
toc;

fprintf('numel(A):%i nnz(A):%i nnz(tA):%i \n', numel(A), nnz(A), nnz(tA)');
end


I first tried a small sparse matrix with 100k elements, 1% sparsity, removing 50% of nonzeros:

A = sprand(1e5,1,0.01); tA = hard_threshold(A, 0.5);
Elapsed time is 0.128991 seconds.
Elapsed time is 0.007644 seconds.
Elapsed time is 0.003038 seconds.
numel(A):100000 nnz(A):995 nnz(tA):489

I next repeated with 1m elements:

A = sprand(1e6,1,0.01); tA = hard_threshold(A, 0.5);
Elapsed time is 15.456836 seconds.
Elapsed time is 0.082908 seconds.
Elapsed time is 0.018396 seconds.
numel(A):1000000 nnz(A):9966 nnz(tA):5019

With 100m elements, excluding the first, slowest, method:

A = sprand(1e8,1,0.01); tA = hard_threshold(A, 0.5);
Elapsed time is 16.405617 seconds.
Elapsed time is 0.259951 seconds.
numel(A):100000000 nnz(A):994845 nnz(tA):498195

The time differential is about the same even when the thresholded matrix is much sparser than the original:

A = sprand(1e8,1,0.01); tA = hard_threshold(A, 0.95);
Elapsed time is 12.980427 seconds.
Elapsed time is 0.238180 seconds.
numel(A):100000000 nnz(A):995090 nnz(tA):49950

The second method fails due to memory constraints for really large sparse matrices:

Error using <
Requested 1000000000x1 (7.5GB) array exceeds maximum array size preference. Creation of arrays greater than this limit may
take a long time and cause MATLAB to become unresponsive. See array size limit or preference panel for more information. Error in hard_threshold (line 10)
tA(abs(tA) < t) = 0;

After excluding the second method, the third method gives:

A = sprand(1e9,1,0.01); tA = hard_threshold(A, 0.5);
Elapsed time is 1.894251 seconds.
numel(A):1000000000 nnz(A):9950069 nnz(tA):4977460

Are there any other approaches that are faster?

# Transforming data to Gaussian

Transforming data to Gaussian using probability integral transform in Matlab:

n = 500;
x=exp(randn(n,1))+(randi(2,[n 1])-1).*(10+3*randn(n,1));
fhat = @(in) sum(x <= in)/n;
Fhat = @(A) arrayfun(fhat,A);
y=Fhat(x); z = icdf('normal',y,0,1);
figure; subplot(1,2,1); hist(x,20); xlabel('x');
subplot(1,2,2); hist(z,20); xlabel('z');


Results:

# Preprocessing and cross-validation

In general, preprocessing should be done inside of cross-validation routine. If you preprocess outside of the cross-validation algorithm (before calling crossval), you will bias the cross-validation results and likely overfit your model. The reason for this is that preprocessing will be based on the ENTIRE set of data but the cross-validation’s validity REQUIRES that the preprocessing be based ONLY on specific subsets of data. Why? Read on:
Cross-validation splits your data up into “n” subsets (lets say 3 for simplicity). Let say you have 12 samples and you’re only doing mean centering as your preprocessing (again, for simplicity). Cross-validation is going to take your 12 samples and split it into 3 groups (4 samples in each group).
In each cycle of the cross-validation, the algorithm leaves out one of those 3 groups (=4 samples=”validation set”) and does both preprocessing and model building from the remaining 8 samples (=”calibration set”). Recall that the preprocessing step here is to calculate the mean of the data and subtract it. Then it applies the preprocessing and model to the 4-sample validation set and looks at the error (and repeats this for each of the 3 sets). Here, applying the preprocessing is to take the mean calculated from the 8 samples and subtract it from the other 4 samples.
That last part is the key to why preprocessing BEFORE crossval is bad: when preprocessing is done INSIDE cross-validation (as it should be), the mean is calculated from the 8 samples that were left in and subtracted from them, and that same 8-sample mean is also subtracted from the 4 samples left out by cross-validation. However, if you mean-center BEFORE cross-validation, the mean is calculated from all 12 samples. The result is that, even though the rules of cross-validation say that the preprocessing (mean) and model are supposed to be calculated from only the calibration set, doing the preprocessing outside of cross-validation means all samples are influencing the preprocessing (mean).
With mean-centering, the effect isn’t as bad as it is for something like GLSW or OSC. These “multivariate filters” are far stronger preprocessing methods and operating on the entire data set can have a significant influence on the covariance (read: can have a much bigger effect of “cheating” and thus overfitting). The one time it doesn’t matter is when the preprocessing methods being done are “row-wise” only – that is, methods that operate on samples independently are not a problem. Methods like smoothing, derivatives, baselining, or normalization (other than MSC when based on the mean) operate on each sample independently and adding or removing samples from the data set has no effect on the others. In fact, to save time, our cross-validation routine recognizes when row-wise operations come first in the preprocessing sequence and does them outside of the cross-validation loop. The only time you can’t do these in advance is when another non-row-wise method happens prior to the row-wise method.

# Resizing nested C++ STL vectors

“Multidimensional” vectors in C++ do not behave like matrices (or higher-order equivalents). Something like:

vector< vector<double> > foo(100, vector<double>(20, 0.0));

will not lay out 100*20 doubles contiguously in memory. Only the bookkeeping info for 100 vector<double>’s will be laid out contiguously — each vector<double> will store its actual data in its own location on the heap. Thus, each vector<double> can have its own size.

This can lead to hard-to-catch bugs:

 foo.resize(300, vector<double>(30, 1.0));

will leave the first 100 vector<double>’s with size 20, filled with 0.0 values, while the new 200 vector<double>’s will have size 30, filled with 1.0 values.

# Sampling from Multivariate Gaussian distribution in Matlab

tl;dr: Don’t use mvnrnd in Matlab for large problems; do it manually instead.

The first improvement uses the Cholesky decomposition, allowing us to sample from a univariate normal distribution. The second improvement uses the Cholesky decomposition of the sparse inverse covariance matrix, not the dense covariance matrix. The third improvement avoids computing the inverse, instead solving a (sparse) system of equations.

 n = 10000; Lambda = gallery('tridiag',n,-0.3,1,-0.3); % sparse tic; x_mvnrnd = mvnrnd(zeros(n,1),inv(Lambda)); toc;

 tic; z = randn(n,1); % univariate random Sigma = inv(Lambda); A = chol(Sigma,'lower'); % sparse x_fromSigma = A*z; % using cholesky of Sigma toc; tic; z = randn(n,1); % univariate random L_Lambda = chol(Lambda,'lower'); % sparse A_fromLambda = (inv(L_Lambda))'; % sparse x_fromLambda = A_fromLambda*z; toc; 

tic; z = randn(n,1); % univariate random L_Lambda = chol(Lambda,'lower'); % sparse x_fromLambda = L_Lambda'\z; toc; 

Results:
 Elapsed time is 4.514641 seconds. Elapsed time is 2.734001 seconds. Elapsed time is 1.740317 seconds. Elapsed time is 0.012431 seconds. 

# Matlab: different colormaps for subplots

I often want different subplots in one Matlab figure to have different colormaps. However, colormap is a figure property, so it’s not trivial, except that it is… with these utilities:

This works for everything except colorbars: http://www.mathworks.com/matlabcentral/fileexchange/7943-freezecolors—unfreezecolors

Post-2010, Matlab refreshes colorbars with each subplot, so you’ll need this to freeze colorbars: http://www.mathworks.com/matlabcentral/fileexchange/24371-colormap-and-colorbar-utilities–feb-2014-