1. Streaming Problems in Astrophysics
Alex Szalay
Institute for Data-Intensive Engineering and Science
The Johns Hopkins University

2. Sloan Digital Sky Survey
“The Cosmic Genome Project”
Started in 1992, finished in 2008
Data is public
2.5 Terapixels of images => 5 Tpx of sky
10 TB of raw data => 100TB processed
0.5 TB catalogs => 35TB in the end
Database and spectrograph built at JHU (SkyServer)
Now SDSS-3/4 data served from JHU

3. Statistical Challenges
Data volume and computing power double every year,
no polynomial algorithm can survive, only N log N
Minimal variance estimators scale as N3, they also optimize on the wrong thing
The problem today is not the statistical variance
systematic errors => optimal subspace filtering (PCA)
We need incremental algorithms, where computing is part of the cost function:
the best estimator in a minute, day, week, year?

4. Randomization and Sampling
In many data sets lots of redundancy
Random subsampling is an obvious choice
Sublinear scaling
Streaming algorithms (linear in the number of items drawn)
How do we sample from highly skewed distributions?
Sample in linear transform space?
Central limit theorem -> approximate Gaussian
Random projections, FFT
Remap PDF onto Gaussian PDF
Compressed sensing

5. Streaming PCA Initialization Incremental updates
Eigensystem of a small, random subset
Truncate at p largest eigenvalues
Incremental updates
Mean and the low-rank A matrix
SVD of A yields new eigensystem
Randomized sublinear algorithm!
Mishin, Budavari, Ahmad and Szalay (2012)

6. Robust PCA PCA minimizes σRMS of the residuals r = y – Py
Quadratic formula: r2 extremely sensitive to outliers
We optimize a robust M-scale σ2 (Maronna 2005)
Implicitly given by
Fits in with the iterative method!

7. Eigenvalues in Streaming PCA
Classic
Robust

8. Cyberbricks 36-node Amdahl cluster using 1200W total
Zotac Atom/ION motherboards
4GB of memory, N330 dual core Atom, 16 GPU cores
Aggregate disk space 148TB (HDD+SSD)
Blazing I/O Performance: 18GB/s
Amdahl number = 1 for under $30K
Using SQL+GPUs for machine learning:
6.4B multidimensional regressions in 5 minutes over 1.2TB
Ported Random Forest module from R to SQL/CUDA
Szalay, Bell, Huang, Terzis, White (Hotpower-09)

9. Numerical Laboratories
Similarities between Turbulence/CFD, N-body, ocean circulation and materials science
On Exascale everything will be a Big Data problem
Memory footprint will be >2PB
With 5M timesteps => 10,000 Exabytes/simulation
Impossible to store
Doing all in-situ limits the scope of science
How can we use streaming ideas to help?

10. Cosmology Simulations
Simulations are becoming an instrument on their own
Millennium DB is the poster child/ success story
Built by Gerard Lemson (now at JHU)
600 registered users, 17.3M queries, 287B rows
Dec 2012 Workshop at MPA: 3 days, 50 people
Data size and scalability
PB data sizes, trillion particles of dark matter
Value added services
Localized
Rendering
Global analytics

11. Halo finding algorithms
2001 SKID Stadel
2001 enhanced BDM Bullock et al.
2001 SUBFIND Springel
2004 MHF Gill, Knebe & Gibson
2004 AdaptaHOP Aubert, Pichon & Colombi
2005 improved DENMAX Weller et al.
2005 VOBOZ Neyrinck et al.
2006 PSB Kim & Park
2006 6DFOF Diemand et al.
2007 subhalo finder Shaw et al.
2007 Ntropy-fofsv Gardner, Connolly & McBride
2009 HSF Maciejewski et al.
2009 LANL finder Habib et al.
2009 AHF Knollmann & Knebe
2010 pHOP Skory et al.
2010 ASOHF Planelles & Quilis
2010 pSO Sutter & Ricker
2010 pFOF Rasera et al.
2010 ORIGAMI Falck et al.
2010 HOT Ascasibar
2010 Rockstar Behroozi
Three pictures here (before FOF, threshold, and final clusters)
1992 DENMAX Gelb & Bertschinger
1995 Adaptive FOF van Kampen et al.
1996 IsoDen Pfitzner & Salmon
1997 BDM Klypin & Holtzman
1998 HOP Eisenstein &Hut
1999 hierarchical FOF Gottloeberg et al.
1974 SO Press & Schechter
1985 FOF Davis et al.
The Halo-Finder Comparison Project
[Knebe et al, 2011]

12. Memory issue
All current halo finders requires to load all the data into memory
Each time snapshot from the simulation with particles will require 12 terabytes of memory
To build a scalable solution we need to develop
an algorithm with sublinear memory usage

13. Streaming Solution: Our goal: haloes ≈ heavy hitters?
Reduce halos finding problem to one of the existing problems in streaming setting
Apply ready-to-use algorithms
haloes ≈ heavy hitters?
To make a reduction to heavy hitters we
need to discretize the space.
Naïve solution is to use 3D mesh:
Each particle now replaced by cell id
Heavy cells represent mass concentration
Grid size is chosen according to typical halo size

14. Count Sketch

15. Streaming Solution: Our goal: haloes ≈ heavy hitters?
Reduce halos finding problem to one of the existing problems in streaming setting
Apply ready-to-use algorithms
haloes ≈ heavy hitters?
To make a reduction to heavy hitters we
need to discretize the space.
Naïve solution is to use 3D mesh:
Each particle now replaced by cell id
Heavy cells represent mass concentration
Grid size is chosen according to typical halo size

16. Count Sketch

17. Memory
Memory is the most significant advantage of applying streaming algorithms.
Dataset size: ~ particles
Any in-memory algorithm: 12 GB
Pick-and-Drop: 30 MB
GPU acceleration
One instance of Pick-and-Drop algorithm can be fully implemented by separate thread of GPU
Count Sketch algorithm have two time-consuming procedures: evaluating the hash functions and updating the queue. The first one can be naively ported to GPU
Zaoxing Liu , Nikita Ivkin , Lin F. Yang , Mark Neyrinck ,
Gerard Lemson, Alexander S. Szalay, Vladimir Braverman, Tamas Budavari, Randal Burns, Xin Wang, IEEE eScience Conference (2015)

18. Summary Large data sets are here
Need new approaches => computable statistics
It is all about systematic errors
Streaming, sampling, robust techniques
Dimensional reduction (PCA, random projections, importance sampling)
More data from fewer telescopes
Large simulations present additional challenges
Time domain data emerging, requiring fast triggers
New paradigm of analyzing large public data sets