Mahout "Samsara" book prerequisites

Q: "... Any tips on prerequisite reading?"

A: To quote the preface of the book:

"Some material assumes an undergraduate level understanding of calculus and on occasion multivariate calculus. The examples are given in Scala and assume familiarity with basic Scala terminology and a minimal notion of functional programming."

Let's elaborate.

There are two sides: the technology side and the math side.

For the technology side, the reader would benefit from some Scala fluency and functional programming. Public Scala material presented on http://scala-lang.org/ gets one a long way, plus of course there's always a longer book by Martin Odersky, et. al., "Programming in Scala".

For the math side, note that we do not explain math. We do not ask ourselves the question "Can I understand it [why it works]?", but rather we ask the question "Can I try it?". A lot of practical research is working exactly just like that -- by asking ourselves the question "What if I try this or that?". So it is not so much about math but rather about math notations, which are for our purposes is pseudocode that we turn into code. Math notations are sufficiently explained in the section A.2; if you have a concrete question beyond what is there, please ask -- we will try to answer.

Beyond notations, it would help to be familiar with the undergraduate level of linear algebra and calculus (after all, we are trying to deal with applied machine learning).

For linear algebra, a good reference is the textbook by Gilbert Strang, "Introduction to Linear Algebra". Algorithms we illustrate at times rely on working knowledge of singular value decomposition (SVD), eigendecomposition, Cholesky decomposition and QR decomposition, as well as Four Fundamental subspaces.

It has been some time since I studied multivariate calculus, so I am not quite sure what the best text is on it these days. I have the "Multivariate Calculus" book by Larson Edwards, which is pretty thorough in my opinion. We do not need all of it though; our book examples touch on very few notions of multivariate calculus -- partial derivatives, gradient, Hessian matrix. As a refresher, perhaps even reading Wikipedia articles on these issues is enough.

Of course when one works on a particular algorithm, she or he needs to read the publication containing the algorithm formulation. That is one of the things that we are trying to demonstrate: how to read and implement algorithm formulations using Mahout "Samsara". We give references to the algorithm publications as appropriate, throughout the book.

-DL

Streaming QR decomposition for MapReduce part III: Collecting Q by Induction

In Part II, as a part of bottom-up divide-and-conquer parallelization strategy, one problem emerged that we need to solve at every node of bottom-up tree:

$$\left(\begin{matrix}\left(\begin{matrix}\mathbf{Q}_{1}\\\mathbf{Q}_{2}\\\vdots\\\mathbf{Q}\end{matrix}\right)\mathbf{R}_{1}\\\left(\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\vdots\\\mathbf{Q}\end{matrix}\right)\mathbf{R}_{2}\\\vdots\\\left(\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\vdots\\\mathbf{Q}_{z}\end{matrix}\right)\mathbf{R}_{n}\end{matrix}\right)=\left(\begin{matrix}\mathbf{\hat{Q}}\\\mathbf{\hat{Q}}\\\vdots\\\mathbf{\hat{Q}}\end{matrix}\right)\mathbf{\hat{R}}$$        (1)       

Let's simplify the problem for a moment (as it turns out, vertical blocking of Q is trivial as they are always produced by application of column-wise Givens operations which we can 'replay' on any Q block).

$$\left(\begin{matrix}\mathbf{Q}_{1}\mathbf{R}_{1}\\\mathbf{Q}_{2}\mathbf{R}_{2}\\\cdots\\\mathbf{Q}_{z}\mathbf{R}_{z}\end{matrix}\right)\Rightarrow\left(\begin{matrix}\mathbf{\hat{Q}}_{1}\\\mathbf{\hat{Q}}_{2}\\\cdots\\\mathbf{\hat{Q}}_{z}\end{matrix}\right)\hat{\mathbf{R}}$$     (2)

Also let's decompose standard Givens QR operation in to two:

1) producing Givens transformation sequence on some input $m\times n$, $m>n$  A as $$\left(\mathbf{G}_{1},\mathbf{G}_{2}...\mathbf{G}_{k}\right)=\mathrm{givens\_qr}\left(\mathbf{A}\right)$$.  I will denote product of all Givens operations obtained this way as $$\prod_{i}\mathbf{G}_{i}\equiv\prod\mathrm{givens\_qr}\left(\mathbf{A}\right)$$.

2) Applying product of Givens operations on input produces already familiar result
\[\left(\prod_{i}\mathbf{G}_{i}\right)^{\top}\mathbf{A}=\left(\begin{matrix}\mathbf{R}\\\mathbf{Z}\end{matrix}\right)\]

from which it follows that thin QR's R can be written as
\[\mathbf{R}=\left[\left(\prod_{i}\mathbf{G}_{i}\right)^{\top}\mathbf{A}\right]\left(1:n,:\right)=\left[\left(\prod\mathrm{givens\_qr}\left(\mathbf{A}\right)\right)^{\top}\mathbf{A}\right]\left(1:n,:\right)\]
using Golub/Van Loan's block notation.

(Golub/Van Loan's block notation in form of A(a₁:a₂, b₁:b₂) means "crop of a matrix A, rows a₁ to a₂ and columns b₁ to b₂". For half- and full-open intervals the bounds are just omitted).

'Thick' Q is produced by $$\mathbf{I}\left(\prod_{i}\mathbf{G}_{i}\right)$$ and hence 'thin Q' is
\[\mathbf{Q}=\left[\mathbf{I}\left(\prod_{i}\mathbf{G}_{i}\right)\right]\left(:,1:n\right)=\left[\mathbf{I}\left(\prod_{i}\mathrm{givens\_qr}\left(\mathbf{A}\right)\right)\right]\left(:,1:n\right),\mathbf{\,\, I}\in\mathbb{R}^{m\times m}.\]

Now that we laid out all notations, let's get to the gist. As I mentioned before, I continue building algorithm by induction. Let's consider case (2) for $$z=2$$. Then it can be demonstrated that the following is the solution as it is equivalent to full Givens QR with a rearranged but legitimate order of Givens operations:

$$\mathbf{\hat{R}}=\left\{\left[\prod\mathrm{givens\_qr}\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)\right]^{\top}\cdot\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)\right\} \left(1:n,:\right)$$,

$$\mathbf{\hat{Q}=}\left[\left(\begin{matrix}\mathbf{Q}_{1} & \mathbf{Z}\\\mathbf{Z} & \mathbf{Q}_{2}\end{matrix}\right)\cdot\prod_{i}\mathrm{givens\_qr}\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)\right]\left(:,1:n\right)$$.

Let's denote function of 2-block computation of $$\mathbf{\hat{R}}$$ as 

$$\mathrm{rhat}\left(\mathbf{R}_{i},\mathbf{R}_{j}\right)=\left\{\left[\prod\mathrm{givens\_qr}\left(\begin{matrix}\mathbf{R}_{i}\\\mathbf{R}_{j}\end{matrix}\right)\right]^{\top}\cdot\left(\begin{matrix}\mathbf{R}_{i}\\\mathbf{R}_{j}\end{matrix}\right)\right\} \left(1:n,:\right)$$.

Still working on the trivial case of induction: Now note that since Givens operations for calculation of $$\mathbf{\hat{Q}}$$ are applied pairwise to the columns of the accumulator matrix, i.e. independently for each row, we can split computation of $$\mathbf{\hat{Q}}$$ and apply it to any combination of vertical blocks of $$\left(\begin{matrix}\mathbf{Q}_{1} & \mathbf{Z}\\\mathbf{Z} & \mathbf{Q}_{2}\end{matrix}\right)$$. Let's say we decide to split it into 2 blocks with as many rows as in Q₁ and Q₂ and get back to block-wise formulas sought for solution of (2):

$$\mathbf{\hat{Q}_{1}=}\left[\left(\begin{matrix}\mathbf{Q}_{1} & \mathbf{Z}\end{matrix}\right)\cdot\prod_{i}\mathrm{givens\_qr}\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)\right]\left(:,1:n\right),$$

$$\mathbf{\hat{Q}_{2}=}\left[\left(\begin{matrix}\mathbf{Z} & \mathbf{Q}_{2}\end{matrix}\right)\cdot\prod_{i}\mathrm{givens\_qr}\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)\right]\left(:,1:n\right).$$      (3)

Let's denote those transformations in more general form as 

$$\mathrm{qhat\_down}\left(\mathbf{Q}_{i},\mathbf{R}_{i},\mathbf{R}_{j}\right)=\left[\left(\begin{matrix}\mathbf{Q}_{i} & \mathbf{Z}\end{matrix}\right)\cdot\prod\mathrm{givens\_qr}\left(\begin{matrix}\mathbf{R}_{i}\\\mathbf{R}_{j}\end{matrix}\right)\right]\left(:,1:n\right)$$

and

$$\mathrm{qhat\_up}\left(\mathbf{Q}_{i},\mathbf{R}_{i},\mathbf{R}_{j}\right)=\left[\left(\begin{matrix}\mathbf{Z} & \mathbf{Q}_{i}\end{matrix}\right)\cdot\prod\mathrm{givens\_qr}\left(\begin{matrix}\mathbf{R}_{j}\\\mathbf{R}_{i}\end{matrix}\right)\right]\left(:,1:n\right)$$

then we can rewrite (3) as

$$\mathbf{\hat{Q}}=\left(\begin{matrix}\mathbf{\hat{Q}}_{1}\\\mathbf{\hat{Q}}_{2}\end{matrix}\right)=\left(\begin{matrix}\mathrm{qhat\_down}\left(\mathbf{Q}_{1},\mathbf{R}_{1},\mathbf{R}_{2}\right)\\\mathrm{qhat\_up}\left(\mathbf{Q}_{2},\mathbf{R}_{2},\mathbf{R}_{1}\right)\end{matrix}\right)$$,

$$\mathbf{\hat{R}}=\mathrm{rhat}\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)$$

This is our solution for (2) of trivial case (z=2).

I will show solution for z=3 and then just will give the final solution without a proof.

Case z=3:

Note that functions qhat_up() and qhat_down() also implicitly produce intermediate rhat() products during their computation, so we want to 'enhance' them to capture that rhat() result as well. We will denote 'evolving' intermediate results Q and R as a sequence $$\left(\mathbf{\tilde{Q}},\mathbf{\tilde{R}}\right)$$:

$$\mathrm{qrhat\_down}\left(\mathbf{Q}_{i},\mathbf{R}_{i},\mathbf{R}_{j}\right)=\left(\mathbf{\tilde{Q}},\mathbf{\tilde{R}}\right)=\left(\mathrm{qhat\_down}\left(\mathbf{Q}_{i},\mathbf{R}_{i},\mathbf{R}_{j}\right),\mathrm{rhat}\left(\mathbf{R}_{i},\mathbf{R}_{j}\right)\right)$$,

$$\mathrm{qrhat\_up}\left(\mathbf{Q}_{i},\mathbf{R}_{i},\mathbf{R}_{j}\right)=\left(\mathbf{\tilde{Q}},\mathbf{\tilde{R}}\right)=\left(\mathrm{qhat\_up}\left(\mathbf{Q}_{i},\mathbf{R}_{i},\mathbf{R}_{j}\right),\mathrm{rhat}\left(\mathbf{R}_{j},\mathbf{R}_{i}\right)\right)$$.

Then, using those notations, we can write solution for (2) z=3 as

$$\mathbf{\hat{Q}}=\left(\begin{matrix}\mathbf{\hat{Q}}_{1}\\\mathbf{\hat{Q}}_{2}\\\mathbf{\hat{Q}}_{3}\end{matrix}\right)=\left(\begin{matrix}\mathrm{qrhat\_down}\left(\mathrm{qrhat\_down}\left(\mathbf{Q}_{1},\mathbf{R}_{1},\mathbf{R}_{2}\right),\mathbf{R}_{3}\right).\mathbf{\tilde{Q}}\\\mathrm{qrhat\_down}\left(\mathrm{qrhat\_up}\left(\mathbf{Q}_{2},\mathbf{R}_{2},\mathbf{R}_{1}\right),\mathbf{R}_{3}\right).\mathbf{\tilde{Q}}\\\mathrm{qrhat\_up}\left(\mathbf{Q}_{3},\mathbf{R}_{3},\mathrm{rhat}\left(\mathbf{R}_{1},\mathbf{R}_{2}\right)\right).\mathbf{\tilde{Q}}\end{matrix}\right)$$.

Note that algorithm for computing $$\mathbf{\hat{Q}}_i$$ requires input of Q_i, iterator[(R₁,R₂...R_z)].

General solution for (2) for any z is built by induction as the following algorithm:

This algorithm is still sub-efficient as for the entire matrix (2) it computes more rhat() operations than needed, and can be optimized to reduce those operations when considering final solution for (1),  but that's probably too many details for this post for now. Actual details are found in code and my working notes.

Finally, a word about how to transform solution for (2) into a solution for (1).

We can rewrite indexes in (1) as

$$\left(\begin{matrix}\left(\begin{matrix}\mathbf{Q}_{1}\\\mathbf{Q}_{2}\\\cdots\\\mathbf{Q}\end{matrix}\right)\mathbf{R}_{1}\\\left(\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\cdots\\\mathbf{Q}\end{matrix}\right)\mathbf{R}_{2}\\\cdots\\\left(\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\cdots\\\mathbf{Q}_{z}\end{matrix}\right)\mathbf{R}_{n}\end{matrix}\right)\equiv\left(\begin{matrix}\left(\begin{matrix}\mathbf{Q}_{11}\\\mathbf{Q}_{12}\\\cdots\\\mathbf{Q}_{1k}\end{matrix}\right)\mathbf{R}_{1}\\\left(\begin{matrix}\mathbf{Q}_{21}\\\mathbf{Q}_{22}\\\cdots\\\mathbf{Q}_{2k}\end{matrix}\right)\mathbf{R}_{2}\\\cdots\\\left(\begin{matrix}\mathbf{Q}_{n1}\\\mathbf{Q}_{n2}\\\cdots\\\mathbf{Q}_{nk}\end{matrix}\right)\mathbf{R}_{n}\end{matrix}\right)$$

And then regroup that into k independent tasks solving

$$\left(\begin{matrix}\mathbf{Q}_{1i}\mathbf{R}_{1}\\\mathbf{Q}_{2i}\mathbf{R}_{2}\\\cdots\\\mathbf{Q}_{zi}\mathbf{R}_{z}\end{matrix}\right)\Rightarrow\left(\begin{matrix}\mathbf{\hat{Q}}_{1i}\\\mathbf{\hat{Q}}_{2i}\\\cdots\\\mathbf{\hat{Q}}_{zi}\end{matrix}\right)\hat{\mathbf{R}}$$,

$$i\in1..k,$$

which can be solved via solution for (2).

That may also be one of  parallelization strategies.

That wasn't so hard, was it? It's possible I am performing a subpar re-tracing of some existing research or method here. But I don't think there's much work about how to fit  QR onto parallel batch machinery.

I hope that some horizons perhaps became a little clearer.

Streaming QR decomposition for MapReduce part II: Bottom-up divide-and-conquer overview

In Part I of "Streaming QR decomposition for MapReduce" I touched a little bit about how bottom-up collection of Q is done. I think I would like to add a couple of figures in attempt to give a little more details and clarifications how it is done.

Collecting Q: Divide-And-Conquer: a tad more details


As I mentioned before, outer QR step is essentially bottom-up n-indegree Divide-And-Conquer algorithm.

\[\mathbf{Y}=\left(\begin{matrix}\cdots\\\mathbf{Y}_{i}\\\mathbf{Y}_{i+1}\\\mathbf{Y}_{i+2}\\\cdots\end{matrix}\right)=\begin{matrix}\cdots\\\left.\begin{matrix}\mathbf{Q} & \mathbf{R}\\\mathbf{Q} & \mathbf{R}\\\cdots & \cdots\\\mathbf{Q} & \mathbf{R}\end{matrix}\right]\\\\\left.\begin{matrix}\mathbf{Q} & \mathbf{R}\\\mathbf{Q} & \mathbf{R}\\\cdots & \cdots\\\mathbf{Q} & \mathbf{R}\end{matrix}\right]\\\\\left.\begin{matrix}\mathbf{Q} & \mathbf{R}\\\mathbf{Q} & \mathbf{R}\\\cdots & \cdots\\\mathbf{Q} & \mathbf{R}\end{matrix}\right]\\\cdots\end{matrix}\Rightarrow\begin{matrix}\cdots\\\left.\begin{matrix}\cdots & \mathbf{\cdots}\\\left.\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\cdots\\\mathbf{Q}\end{matrix}\right] & \mathbf{R}\\\\\left.\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\cdots\\\mathbf{Q}\end{matrix}\right] & \mathbf{R}\\\\\left.\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\cdots\\\mathbf{Q}\end{matrix}\right] & \mathbf{R}\\\cdots & \cdots\end{matrix}\right]\\\cdots\end{matrix}\Rightarrow\cdots\Rightarrow\begin{matrix}\left.\begin{matrix}\mathbf{\cdots}\\\mathbf{\hat{Q}}\\\mathbf{\hat{Q}}\\\cdots\\\mathbf{\hat{Q}}\\\\\mathbf{\hat{Q}}\\\mathbf{\hat{Q}}\\\cdots\\\mathbf{\hat{Q}}\\\\\mathbf{\hat{Q}}\\\mathbf{\hat{Q}}\\\cdots\\\mathbf{\hat{Q}}\\\cdots\end{matrix}\right] & \hat{\mathbf{R}}\end{matrix}\]

 The way generalized Givens thin QR works, it applies a number of Givens transformations on a Q-accumulator matrix given some initial tall matrix (in this case, Y blocks) until the tall matrix is reduced to a form of $\left(\begin{matrix}\mathbf{R}\\\mathbf{Z}\end{matrix}\right)$ , where Z has all zero elements and R is an upper-triangular.   Initial value for Q-accumulator matrix is chosen as I (square identity matrix). To tranform result to a 'thin' QR result, only first n columns of accumulator are taken (that becomes Q of the decomposition) and R part is taken as the second matrix of the thin decomposition.

In our case, in order to enable Givens recursive use, we generalize standard Givens algorithm by making it  accept two parameters: pre-existing 'thick qr' Q accumulator instead of I (in reality only shifting 'thin' Q accumulator is sufficient) and the tall input matrix as second input.

Bottom-up divide-and-conquer algorithm hence can be described as follows:

• First, we split all input (Y) into vertical blocks (see illustration above). For each block, we compute slightly modified version of standard Givens QR which is optimized for sequential (streaming) access to the tall input block elements without running into big memory requirements. The output of this step is bunch of (Q,R) pairs (each pair corresponds to the original block). For benefit of subsequent step, we will consider this block data as ({Q}, R) where sequence of Q blocks is denoted as {Q} and contains only one matrix. (Actually math stuff normally uses parenthesis () to distinguish sequences from sets, but I feel using {} notation is much more expressive in this case).
  
  
• Second, we group the input of form {Q}, R into new groups, each group thus would be denoted as {{Q},R} such that number of R in each group doesn't exceed certain maximum bottom-up indegree limit N (usually 1000 for 1G RAM solvers).
  
  
• Third, for each group formed in step 2 above, we run row-wise Givens solver, which produces ("merges")  a valid Givens solution for  each group {{Q},R} → {Q},R. Essentually this solves "QR reduction" at nodes of divide-and-conquer:
  
  
  ,
  which is essentially equivalent to solving a number of individual block-wise QR decompositions over a one ore more initial blocks of Y into single one
  .
  This algorithm is called 'compute QHatSequence2' (I think) in my detailed notes and is the one that builds by induction. Hat sign is to denote final QR blocks. I plan to discuss details of that algorithm in "part III" of this blog.
  
  
• Fourth, we consider each result of step 3 to form a sequence again and restart form step 2 and repeat it from there unless the number of groups is 1, which would produce our final Q blocks and single R as a result.

Parallelization Strategy

1. We can consider the iterations above from the point of view of individual block Y_i as a set of isolated parallel steps. We can view the entire computation as a series of independent computations over Q_i,{R₁...R_n}. It also turns out we always consume the sequence of R in the same order and never have to load more than one upper triangular matrix in the memory, so the parameters of such algorithms can actually be [Q_i, iterator{R₁...R_n}]. The algorithm produces new version of Q block, and the last solver in a group would produce a new R item for the next step R-sequence (as can be demonstrated further on). Then solver is reloaded with next R sequence and runs again (until we are left with just one R). That would be next map-only job run but each run reduces total number of Rs thousands of times. so 3 map-only runs can handle 1 billion blocks (besides the last run is really combined with the next step, computation of $\mathbf{B}^{\top}$, so we save at least one iteration setup overhead here.
2. Another small enhancement is that it turns out each Q computation doesn't need entire {R₁...R_n}. Let's rewrite Q block indexes into 2-variable index to reflect their group and subgroup indices:\[\left(\begin{matrix}\left(\begin{matrix}\mathbf{Q}_{1}\\\mathbf{Q}_{2}\\\cdots\\\mathbf{Q}\end{matrix}\right)\mathbf{R}_{1}\\\left(\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\cdots\\\mathbf{Q}\end{matrix}\right)\mathbf{R}_{2}\\\cdots\\\left(\begin{matrix}\mathbf{Q}\\\mathbf{Q}\\\cdots\\\mathbf{Q}_{z}\end{matrix}\right)\mathbf{R}_{n}\end{matrix}\right)\equiv\left(\begin{matrix}\left(\begin{matrix}\mathbf{Q}_{11}\\\mathbf{Q}_{12}\\\cdots\\\mathbf{Q}_{1k}\end{matrix}\right)\mathbf{R}_{1}\\\left(\begin{matrix}\mathbf{Q}_{21}\\\mathbf{Q}_{22}\\\cdots\\\mathbf{Q}_{2k}\end{matrix}\right)\mathbf{R}_{2}\\\cdots\\\left(\begin{matrix}\mathbf{Q}_{n1}\\\mathbf{Q}_{n2}\\\cdots\\\mathbf{Q}_{nk}\end{matrix}\right)\mathbf{R}_{n}\end{matrix}\right).\] It turns out that computation over Q_1i and Q_2i  requires entire {R₁...R_n} but computation over Q_3i requires $\left\{ \mathrm{GivensQR}\left[\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)\right].\mathbf{R},\mathbf{R}_{3}...\mathbf{R}_{n}\right\}$, i.e only n-1 upper-triangulars. Going on, сomputation over Q_4i requires sequence $\left\{ \mathrm{GivensQR}\left[\left(\begin{matrix}\mathrm{GivensQR}\left[\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\end{matrix}\right)\right].\mathbf{R}\\\mathbf{R}_{3}\end{matrix}\right)\right].\mathbf{R},\mathbf{R}_{4}...\mathbf{R}_{n}\right\}$. And so on, with the Q_n requiring only two upper-triangular arguments. That means that we quite legally can split computation into at most k independent jobs, each i-th parallel job computing blocks Q_1i, Q_2i...Q_ni in sequence while also reducing R sequence per above. To aid parallelization even more, we can actually choose k to be whatever we want: inside the algorithm, matrix $\left(\begin{matrix}\mathbf{Q}_{i1}\\\mathbf{Q}_{i2}\\\cdots\\\mathbf{Q}_{ik}\end{matrix}\right)$ is only ever transformed by applying column-wise Givens operations and horizontal shifts. Hence, it doesn't matter how it is being split into blocks, we can regroup rows there in any way to come up with a degree of parallelism k we are comfortable with.

Looks pretty complicated, huh? I guess it's not most complicated part yet though. Besides, I haven't implemented bottom-up approach 100% yet. I only implemented 2-step hierarchy (i.e. we can have n×n Y blocks initially only). I think it is kind of enough for what my company does, but it can be worked on to insert more steps in between to enable as many blocks as we want.

We've just started on the path uphill.

Streaming QR decomposition for MapReduce part I: Induction, Divide-and-conquer and other aging but still neat goodies

One of most prominent challenges in my stochastic SVD contribution to Mahout was low-memory parallelizable implementation of a QR decomposition. Although current implementation of QR decomposition is rather tightly coupled with and rather specific for the SSVD method, it can be re-worked to be a massive scale QR method standing on its own.

Briefly, my solution is equivalent to one giant row-wise Givens solver. The trick is to legally reorder Givens operations and distribute their computations to parallel running pipelines so that most of the work doesn't require bulk data recombination.

There are several design patterns and principles used.

Streaming preprocessors

One principle I used was nested streaming preprocessors. Imagine having a bunch of lexical preprocessors working on a program source. First preprocessor strips out comments, second preprocessor parses grammar alphabet, etc. This pattern allows to cram a lot of logic in one sequential processing pass without having to load 'the whole thing' into memory.

So I tried to follow the same principle in matrix processing pipeline. First step is to produce matrix Y. We consume matrix A but we never even form so much as a complete row of A in memory. This is one of the enhancements, being able to consume A in element-by-element fashion (only non-zero elements for sparse data). We can accumulate row of Y as (k+p) long dot-product accumulator and that's all the memory this particular preprocessor needs. Once we finished with all alements in row of A,  the dot-product accumulator contains final row of Y which is then passed on onto first step of QR pipeline.

We start off QR pipeline by dividing matrix Y into z horizontal blocks.

Again, Y blocking is only conceptual, as actual blocks are never formed in one place. To put things in perspective, each map task works with one or more Y blocks.

QR pipeline works on Y rows, row by row, by applying bottom-up ordered Givens operations until it transforms Y block into the following tall matrix:

\[\mathbf{Y}_{i}\rightarrow\mathrm{GivensQR}\rightarrow\left(\begin{matrix}\times & \times & \cdots & \times & \times\\0 & \times & \cdots & \times & \times\\\vdots & \vdots & \ddots & \vdots & \vdots\\0 & 0 & \cdots & \times & \times\\0 & 0 & \cdots & 0 & \times\\\\0 & 0 & 0 & 0 & 0\\\vdots & \vdots & \vdots & \vdots & \vdots\\0 & 0 & 0 & 0 & 0\end{matrix}\right)=\left(\begin{matrix}\mathbf{R}_{i}\\\mathbf{Z}\end{matrix}\right)\]

This tall matrix is represented with an upper-triangular R matrix, which is (k+p)×(k+p) sitting on the top of tall zero matrix Z. The R matrix is the one from thin QR decomposition, and we can get rid of Z matrix as having no information. 

In fact, our QR preprocessor only keeps R-sized (k+p)×(k+p) buffer which "slides" up until it ends up holding R. Each additional iteration "builds" 1 row of Y on top of buffer, turning it into something resembling upper Hessenberg form, and then second part of iteration eliminates "subdiagonal" by applying Givens iterations and turning it back to upper-triangular. Last row is then completely zeroed and thrown away (or rather, reused as buffer for Y row so we don't incur java GC thrashing too much).

Divide and Conquer

The next step is to merge blockwise QR results into single final QR result. in order to do that, we can stack up individual blockwise R matrices one on top of another and apply same strategy, namely, selectively reordered Givens set until we end up with R/Z result again:

\[\left(\begin{matrix}\mathbf{R}_{1}\\\mathbf{R}_{2}\\\cdots\\\mathbf{R}_{n}\end{matrix}\right)\rightarrow\mathrm{GivensQR}\rightarrow\left(\begin{matrix}\mathbf{R}\\\mathbf{Z}\end{matrix}\right)\]

The result of this is final QR decomposition (or at least R part of it). 

The next notion is that we can apply those two steps recursively in bottom-up divide-and-conquer fashion to merge as many intermediate QR blocks as we want. 

This is basically all that happens in MapReduce QR: mappers are running one or more QR blocks, optionally merging them; and then another pass goes up one level on divide-and-conquer hierarchy tree, and then the whole routine repeats until we are left with just one R.

Induction for collecting Q 

Producing (and merging) Q is somewhat more complicated to describe than producing R.   But the whole approach of bottom-up ~1000-indegree divide-and-conquer (or perhaps merge-and-conquer :) is the same as described in the previous section even as we evolve Q blocks thru the merge hierarchy. The algorithm for individual step of Divide-and-Conquer bottom-up iteration there builds by induction and is described in my notes. The challenge is to stream individual Q blocks through series of Givens rotations that eventually it would have to go through in order to become a block of the final Q matrix. The algorithm builds for a trivial case of induction for a single bottom-up divide-and-conquer step of indegree 2 and 3 and then proceeds with building general case of indegree n. Memory requirements are to hold one Q block in addition to R sequence during Q merging. If one merge-up is one map-only MapReduce jobs, then just two map-only MR jobs can merge up 1,000,000 blocks or more than a billion rows (assuming Y blocks can be at least 1000 rows high -- but they can be much more in practice).

Turns out the number of Givens operations that we need to apply at each evolution of Q-block can be significantly smaller than the number elements of Q block we need to apply them to because it doesn't depend on the height of the Q-block but only proportional to indegree of the bottom-up divide-and-conquer pass and also ~(k+p)/2. That basically means that we could perform each pass of divide-and-conquer as a map-only pass with Givens operations being a side file information. (actually Givens operations are produced from that stack of Rs shown above, but they are different for each Q block, so that stack of Rs is the side information and Givens are produced on-the-fly). There are also some optimization techniques to transform (reduce) R-stack as we progress thru Q blocks to avoid unnecessary computations, so at the end we end up with only one R in the stack which also happens to be final R for the local step of divide-and-conquer bottom-up process. We take the final R and use that in subsequent divide-and-conquer merge-ups.

In practice in Stochastic SVD we piggyback on the mappers of the second job to finish QR merging -- and then pass it on onto whatever second job is really supposed to do (computing matrix B). That practically would allow to achieve QR per my previous estimate of 1 billion or more rows in mappers with 1Gb RAM with what seems like practically a single map-only pass over input data. (A patch enabling preprocessing input vectors is not part of Mahout patch -- not yet at least, so Mahout version may have significantly higher RAM requirements and longer running time due to GC thrashing than my github branches  if the input contains rather wide and dense vector data.)

So, to recap, the major design principles in MR-based QR were: streaming preprocessors; recursive bottom-up divide-and-conquer; design by induction. Big thanks to Udi Manber and his books on algorithm design basics that got pretty well imprinted in the inner side of my skull.

My perception of LSI