Tensions in Practice: Pattern sparsity in tension with direct coordinate access¶
Two-dimensional vectors · lossless representation
The vector (1, 1) needs two nonzero coefficients in the usual horizontal and vertical axes. In a basis whose first vector is (1, 1), it needs only one. Nothing was discarded: both bases still describe the entire two-dimensional space. The trade appears when a different vector or operation is considered; a convenient basis for one pattern can complicate another.
Read ordinary components directly
Keep the first stored coefficient equal to the vector’s first ordinary component.
Expose recurring patterns sparsely
Represent the two diagonal patterns with one nonzero coefficient apiece.
Why these aims pull against each other
The count of nonzero coefficients depends on the basis. Making diagonal patterns sparse makes the ordinary-axis vector (1, 0) dense and requires decoding ordinary components.
Choose an arrangement to see what changes and what remains difficult.
Finite illustrative comparisons. Text states carry the meaning; color is not a measured score or universal preference.
What this choice protects
What it costs
When it fits
Compare the arrangements
Ordinary axes
Use basis (1, 0) and (0, 1). Stored coefficients (a, b) reconstruct the vector directly as (a, b).
| Coefficients | Nonzero | First value | |
|---|---|---|---|
| (1, 1) | (1, 1) | 2 | 1 |
| (1, −1) | (1, −1) | 2 | 1 |
| (1, 0) | (1, 0) | 1 | 1 |
- What it protects
- The first component is read directly; the axis vector (1, 0) has one nonzero coefficient.
- What it costs
- Both diagonal patterns in the table require two nonzero coefficients.
- When it fits
- Ordinary component access or axis-aligned patterns dominate the work.
Illustration note: The finite setting and values are editorial assumptions, not measured effects or recommended operating settings. “Nonzero” counts coefficients, not bytes or runtime. All arithmetic is exact.
Diagonal axes
Use basis (1, 1) and (1, −1). Coefficients (a, b) reconstruct the ordinary vector as (a+b, a−b).
| Coefficients | Nonzero | First value | |
|---|---|---|---|
| (1, 1) | (1, 0) | 1 | 1 |
| (1, −1) | (0, 1) | 1 | 1 |
| (1, 0) | (½, ½) | 2 | 1 |
- What it protects
- Each shown diagonal pattern has one nonzero coefficient.
- What it costs
- The axis vector needs two; reading the first ordinary component now requires a+b, and the decoding basis must be known.
- When it fits
- Diagonal patterns recur enough to justify the conversion and decoding convention.
Illustration note: The finite setting and values are editorial assumptions, not measured effects or recommended operating settings. Coefficients are a=(x+y)/2 and b=(x−y)/2. The table retains both basis directions; zeros do not mean a dimension has been deleted.
What this illustration does—and does not—establish
Basis: Sparsity in One Basis versus Density in Another (measurement) supplies basis-relative sparsity; its explicit distinction from reduction bounds the complete two-coordinate editorial example.
- Both representations are invertible and lossless for all real pairs.
- No fitted model, denoising, dimension reduction or net compression ratio is claimed.
- The basis that simplifies these patterns is not best for every vector or operation.
Source entries
Basis
Basis: Sparsity in One Basis versus Density in Another (measurement) supplies the conflict examined here.
Sparsity in One Basis versus Density in Another (measurement)
An element can be complicated (dense) in one basis and simple (sparse) in another, licensing compression and recovery — but sparsity is basis-relative, so the wrong basis hides the structure entirely.
What It Is Not
- Not dimensionality_reduction. Reduction discards dimensions to approximate a space in fewer; a basis is an *exact, lossless* minimal generating set for the full space. Choosing a data-derived basis can *enable* reduction, but the basis itself loses nothing.