Random projection dimensionality reducer

About this tool

Random projection is the cheapest useful dimensionality reduction there is. Instead of studying your data to find good axes — the way PCA computes eigenvectors — it multiplies the table by a matrix of random numbers and simply relies on a theorem: the Johnson–Lindenstrauss lemma says that a random linear map into enough dimensions keeps every pairwise distance almost unchanged, and that "enough" depends only on how many points you have, never on how wide the table was.

That makes it the tool of choice when a table is far too wide to decompose — thousands of term-frequency columns, embedding dimensions, sensor channels — and when what you need preserved is distances (for nearest-neighbour search, clustering, duplicate detection) rather than variance.

Paste your matrix and this reducer does the whole thing in the browser:

• draws a k × columns random matrix from one of four families — Gaussian, sparse, Achlioptas or Rademacher — each scaled so that distances are preserved in expectation;
• projects every row onto it, giving k new coordinates named RP1, RP2, …;
• measures what actually happened: it recomputes pairwise row distances before and after and reports the mean, median and maximum distortion, the mean ratio, and how many sampled pairs landed inside your ±eps;
• tabulates the Johnson–Lindenstrauss minimum dimension for several tolerances, so the trade-off between eps and k is visible in the same result.

Nothing is uploaded. The random numbers come from a fixed, portable integer stream rather than a platform generator, so a given seed reproduces exactly the same projection on this page, on the command line and in chat.

Worked example — eight channels down to three

Paste six observations of eight channels (the first row is read as a header of column names) and set Target dimensions to 3:

ch1,ch2,ch3,ch4,ch5,ch6,ch7,ch8
1,2,3,4,5,6,7,8
8,7,6,5,4,3,2,1
2,4,6,8,10,12,14,16
0,1,0,1,0,1,0,1
5,5,5,5,5,5,5,5
9,1,8,2,7,3,6,4

The report opens with the settings and the honest verdict on quality:

Random projection: 6 rows × 8 columns → 3 dimensions (37.5000% of the input width)
  method        gaussian — dense matrix, entries drawn from N(0, 1/k)
  target dims   3 (set explicitly)
  density       100.0000% non-zero entries
  seed          42
  input columns ch1, ch2, ch3, ch4, ch5, ch6, ch7, ch8

Distance preservation (15 of 15 row pairs measured):
  mean distortion     22.4111%
  median distortion   21.1289%
  max distortion      42.5474%
  mean ratio          1.051128  (projected ÷ original distance)
  within ±10.0000%  3 of 15 pairs (20.0000%)

and ends with the projected coordinates:

Projected data (first 6 of 6 rows):
  row           RP1           RP2           RP3
    1    -17.064939     -0.160032     -8.288287
    2    -13.495953      4.349463     -1.436021
    3    -34.129877     -0.320064    -16.576574
    4     -1.341949     -0.556681     -0.113729
    5    -16.978273      2.327462     -5.402393
    6    -20.080918      7.953206     -6.799878

A 22% mean distortion is bad, and the tool says so rather than hiding it — three dimensions is simply not enough for a guarantee. The Johnson–Lindenstrauss block in the same report explains why: even six points want k ≥ 1535 for a ±10% embedding. Random projection is a large-k, wide-data method, and a six-row toy is the case where it works worst. It is still the right example to start from, because every number above is exactly reproducible — same seed, same result, on every surface.

What it looks like when there is room to work

On a table of 60 rows and 200 columns of random values, raising k tightens the embedding in exactly the way the lemma predicts:

The sparse family at the same k = 64 lands at 7.51% mean distortion while touching roughly one entry in fourteen of the matrix — slightly noisier, much less work. That trade is the whole reason the sparse variants exist.

Choosing the target dimension

• A number (3, 64, 128) sets k directly. This is the usual choice: pick the width your downstream index or model wants, then read the distortion figures to see what it cost.
• A percentage (25%) keeps that share of the input width, which is handy when the same setting is applied to tables of different widths.
• auto derives k from the Johnson–Lindenstrauss bound at your eps, then clamps it to the number of columns you actually have. On a small paste that clamp bites immediately — the bound wants more dimensions than the data has — and the report says so.

Reading the output formats

text is the report above and prints the first 20 projected rows. csv emits every projected row as row,RP1,RP2,… so it can go straight into a plot or another tool. json returns the whole result: every row, the diagnostics, and the projection matrix. matrix returns just the k × columns projection matrix as CSV — that is what you keep if you need to project new rows onto the same axes later, since the matrix plus the seed is the entire model.

Limits and edge cases

• Shape. At least 2 rows and 2 columns; up to 2,000 rows, 1,000 columns and 200,000 cells. Target dimensions run from 1 to 256.
• Separators. Commas, tabs, semicolons, pipes and runs of spaces all work, so a spreadsheet paste, a CSV or a fixed-width dump is fine. Blank lines are skipped.
• Header detection. The first row is read as column names only when its tokens are not all numeric. Column names are echoed in the report and used as the header of the matrix output.
• Every cell must be a finite number. A missing, blank or non-numeric cell is a located error (row 4, column 2: 'n/a' is not a finite number), never a silent zero, and a ragged row is rejected with both row lengths.
• Diagnostics are sampled. All pairs are measured when there are 200 rows or fewer; above that 20,000 row pairs are sampled deterministically from the seed. Pairs of identical rows have zero original distance, so no ratio exists for them — they are counted as skipped.
• Density applies only to sparse and achlioptas. Setting it on a dense family (gaussian, rademacher) is an error rather than a silently ignored field.
• Not variance-optimal. For the same k, PCA finds better axes; random projection wins on speed and on tables too wide to decompose, and its guarantee is about distances, not variance.
• Rounding. All reported numbers are rounded to 6 decimal places.

FAQ