Separated, and narrower

A line at the real kernel’s slope settled how the rank of an oscillatory interaction block grows once the two clusters are many wavelengths across. The block is the cos(κr)/r kernel sampled between two equal parallel segments at the separation a hierarchical matrix calls admissible. Past a Fresnel number $F = \kappa ab/(2\pi D)$ of about 16 its rank at $10^{-8}$ is a straight line in $F$, with a slope of 1.4 to 1.6. That was below the prediction of two, one family of waves for each direction a real kernel travels in, and the essay found why. At each point of the target the source is seen over a band of directions; the block’s columns oscillate there at local frequencies $\kappa\sin\theta$ across that band. A complex kernel $e^{i\kappa r}$ needs one band, a real one needs the band and its reflection through zero — and when the segments face each other the two overlap in the middle of the target. The rank counts their union, $1.40F$, not their sum, $1.89F$.

That essay ended by naming the geometry that should remove the overlap. “For two equal segments offset along their own line so that the target sees the source entirely to one side, the outgoing and incoming bands never overlap. The prediction with a sign is that such a block’s rank, at the same $F$, follows twice the complex count — a slope near 1.9 rather than 1.4 — and that the change from one slope to the other happens as the offset carries the band past zero, at an offset of about half the segment length.”

The bands separate as predicted. The slope does not follow them, and the reason is in the same picture the prediction drew.

The same block, slid sideways #

Everything is the earlier construction: segments of equal length $a$ at the admissible separation $D = 2a$, twelve samples a wavelength, wavenumber 40, ranks counted from a randomised range finder at $10^{-8}$. The only change is that the source segment is shifted along its own line by $s\,a$, for $s$ from 0 to 2, so that at $s = 1$ the source’s near end is level with the target’s far end and every line from target to source leans the same way. The Fresnel number keeps its definition through the perpendicular separation, so a given $F$ is the same pair of segments at every offset.

The rank’s slope is 1.54 centred, 1.58 at a quarter of a length, 1.67 at a half and three quarters, 1.58 again at one length, 1.25 at one and a half and 0.92 at two. It never comes near 1.89. It rises by a twelfth between centred and its peak and then falls below where it started. The union’s slope — the phase-space count of the earlier essay, computed for each offset — runs below it at every offset by 0.12 to 0.18, and has the same shape.

The band and its reflection #

The phase-space picture shows what the prediction saw and what it did not.

Facing, the band runs from $-0.24$ to $0.24$ at the target’s centre, crossing zero, and its reflection covers almost the same strip; over the middle of the target the two coincide, and the union is three quarters of their sum. Shifted by a length, the band sits wholly above zero — from 0.24 to 0.60 at the target’s centre — and its reflection wholly below, and the union is exactly the sum, twice the band. That much of the prediction is exactly true.

What the picture also shows is the band’s width. Facing, it spans almost half the range of $\sin\theta$; shifted, it is thinner, because the source is seen obliquely: $\sin\theta = t/\sqrt{t^2 + D^2}$ is steep near zero and flat towards one, and a segment of the same length seen from the side subtends a smaller range of sines than one seen head-on. The band’s area — the complex kernel’s count of independent waves — falls from 0.94 per unit of $F$ facing to 0.71 at one length and 0.37 at two.

The narrowing is a matter of the sine’s slope. A source point at horizontal offset $t$ from a target point, at height $D$, is seen at $\sin\theta = t/\sqrt{t^2 + D^2}$, whose rate of change with $t$ is $D^2/(t^2 + D^2)^{3/2}$: largest at $t = 0$ and falling as the point moves off to the side. A source facing the target is spread around $t = 0$, where each unit of its length buys the most directions; a source a length to the side is spread over $t$ from $a/2$ to $3a/2$ at the target’s centre, where the sine is flattening. The numbers at the centre of the target bear it out: facing, the source covers sines from $-0.24$ to $0.24$, a width of 0.49; shifted by a length, from 0.24 to 0.60, a width of 0.36 — three quarters as wide, which is the ratio of the two blocks’ band areas, 0.71 to 0.94, to within the variation along the target.

The overlap was worth what the obliquity costs #

The three counts tell the whole story between them. Twice the band starts at 1.89 and falls steadily as the source slides sideways. The union starts at 1.40, below twice the band by the overlap; it rises a little as the overlap shrinks, to 1.54 at three quarters of a length; and at one length, where the overlap is gone, it meets twice the band at 1.42 — almost exactly the facing union’s 1.40. Beyond that the two are the same curve, falling. Removing the overlap gains back what the overlap cost, and moving the source far enough to remove it costs, in obliquity, almost exactly as much.

The prediction’s error was to hold the band fixed while moving the source. In the earlier essay’s phrasing, the facing block’s discount from 1.89 to 1.40 was “the overlap”; it is equally true to say that a facing source is seen at the steepest part of the sine, where a given length of segment buys the most directions. Sliding it sideways trades one advantage for the other, and at one length the trade is nearly even.

The rank follows the count #

At one length the rank is 31, 46, 60, 74 and 98 at $F$ = 8, 16, 24, 32 and 48, against the facing block’s 25, 39, 51, 64 and 88: a little more, at every $F$, by 6 to 10 columns — not the 20 to 30 more that a slope of 1.9 would have given by $F = 48$. The union count is 11.4 to 68.4, and the rank sits above it by a margin that is nearly fixed in columns rather than in proportion: 21 at $F = 48$ facing, 30 at one length. Turn the dial to two lengths and the rank falls below the facing block’s at every $F$ from 16: 35, 44, 51 and 66 against 39, 51, 64 and 88. A block seen from far enough to the side is cheaper than a facing one, not dearer.

In proportion the offset matters most where the blocks are smallest. At $F = 8$ the block a length to the side needs 31 columns to the facing block’s 25, a quarter more; at $F = 48$, 98 to 88, a ninth more. The difference in columns stays between 6 and 10 while the ranks triple, which is what a margin that depends on the geometry but hardly on $F$ would do, and it means that for the small, near-diagonal blocks of a hierarchical matrix the side-on geometry is relatively dearer than the slope alone suggests.

The ratio of the rank to the union count falls with $F$ at every offset — from 2.2 to 1.31 facing, from 2.7 to 1.43 at one length, from 4.4 to 1.86 at two — which is the earlier essays’ finding that the count is the slope and not the rank: the rank is the count plus a margin that grows only slowly, the edge of the band rounded off at the tolerance. A rank that is a number of digits measured that margin’s growth with the tolerance on a smooth kernel; here it is 21 to 32 columns at $F = 48$, and larger the further the source sits to the side, because a narrow band at a grazing angle has proportionally more edge.

Turning the source instead #

If the rank’s slope is the area of the directions’ union, it should follow that area under any change of geometry, not only a slide. The cleanest other change is a turn: keep the source’s centre where it was, facing the target’s, and rotate the segment about it, so that its band of directions stays centred on zero and only its width changes. Measured the same way, at 30, 60 and 90 degrees, the rank at $F = 24$ and 48 is 51 and 88 facing, 52 and 87 at 30 degrees, 44 and 72 at 60, and 18 and 26 at 90, where the source points straight at the target and subtends almost nothing. The slopes are 1.54, 1.46, 1.17 and 0.33; the union count’s slopes are 1.40, 1.30, 0.88 and 0.25. A turn changes nothing about the overlap, which stays complete at the centre of the target, and everything about the width, and the rank follows the width. The margin over the count is 0.15, 0.16, 0.29 and 0.08: larger than for the slide at 60 degrees, where the band is narrowed unevenly along the target, but the same size, a small fraction of the slope.

So two geometries that move the rank in quite different ways — a slide that separates the bands and narrows them, and a turn that only narrows them — are both read correctly by one number computed without a single singular value. A block nobody can call sparse was the essay that first separated a block’s few singular values that matter from the many it has; the union count is a way to know how many matter from the geometry before the block is formed, and the test that costs what it saves is the reminder that any test deciding a partition has to be cheaper than the compression it guides. This one is a quadrature over the target, a few thousand evaluations of a sine.

At other tolerances #

The slopes above are at $10^{-8}$, and the earlier essays found the facing slope nearly independent of the tolerance; offset blocks were not checked there, so they were checked here, from the same sketched spectra. At $10^{-4}$ the slopes from centred to two lengths are 1.46, 1.46, 1.58, 1.58, 1.50, 1.13 and 0.83; at $10^{-6}$, 1.50, 1.58, 1.58, 1.63, 1.54, 1.21 and 0.92. The shape is the same at every tolerance — a small rise to a peak between a half and three quarters, then a fall below the facing value — and the whole curve moves up a little as the tolerance tightens, because a tighter tolerance keeps more of the band’s rounded edge. At $10^{-10}$, the tightest the sketch resolves, the peak reaches 1.79 at half a length, the nearest any measurement here comes to the predicted 1.89, and it is reached where the bands still partly overlap rather than where they have separated.

That last point is the one worth keeping. Whatever the tolerance, the steepest block is not the one with separated bands; it is the one with bands that have mostly separated while the source is still seen nearly head-on. The prediction looked for the doubling at the far end of the slide, and the far end is where the slide costs most.

What the slope is a count of #

The measurement replaces the prediction’s dichotomy — union or sum — with a single quantity that covers both: the area of the union of the band and its reflection over the target, in units of $2\pi$. Facing, the union is less than the sum because the two overlap; sideways, the sum is less than it was because the band narrows; and the rank’s slope follows the union through both regimes to within 0.2 at every offset measured. The steepest geometry is neither of the prediction’s two cases but the one between them, half to three quarters of a length off-centre, where the overlap is mostly gone and the obliquity has not yet set in.

For a hierarchical matrix the practical reading is about which blocks are expensive. Which pairs are allowed to be small and the admissibility rule decide which blocks are compressed by their geometry alone; the smaller cluster sets the rank found the cost following the smaller cluster when the two differ in size. Offset is a third geometric variable, and it moves the cost by a factor of less than two over the range a partition produces: a block slightly off-centre is the dearest, a block far to the side is the cheapest, and none is dearer than the facing block by more than a sixth — 101 columns at $F = 48$ at the steepest offset against 88 facing. The kernel with nothing to compress found the oscillatory kernel’s rank growing without limit as the clusters grow at fixed wavelength; offset changes the slope of that growth, not the fact of it.

The sampling, too, deserves a word, since it is what makes the rank a property of the geometry. The size the rank does not notice found a well-separated block’s rank independent of how finely it is sampled once the samples resolve the kernel; here every block is sampled at twelve points a wavelength, so the number of rows grows with $F$ while the rank grows with the union count, and the ratio of the two — the compression a hierarchical matrix gets from the block — improves with $F$ at every offset, from about an eighth of the rows at $F = 8$ to a thirteenth at 48.

What one pair of segments does not show #

The segments are collinear-parallel, equal and straight; real clusters are surfaces with every orientation, and for them the band of directions is two-dimensional and its reflection’s overlap is a matter of solid angle. The offsets run along the segments’ own line at a fixed perpendicular separation, so the source also moves further away as it slides, and that added distance is part of the obliquity measured here rather than something separated from it. The slopes are read between $F = 24$ and 48 only; the earlier essay found the facing slope settling from 16 on, and at two lengths the curve may still be bending at 48. And the ranks are at one tolerance, $10^{-8}$; the earlier essays found the slope nearly independent of it, and that is assumed here, not checked.

Still open: a source turned, and the steepest offset at scale #

A curved source. An arc of the same length as the source segment, bent towards the target, sees the target over a wider range of directions at each of its points than a straight segment does. The prediction with a sign is that an arc of 60 degrees facing the target has a rank slope at least a tenth above the straight segment’s from $F = 24$ to 48, and that its union count, computed from the arc’s two ends as here, underestimates that slope by more than it does for any straight source — because a curved source’s band is no longer bounded by its endpoints’ directions alone.

The steepest offset at larger $F$. The rank’s slope peaks between half and three quarters of a length. The prediction is that the peak stays in that range from $F = 48$ to 96, that its height there converges to the union’s 1.54 plus a margin that falls with $F$, and that the facing and one-length slopes converge to within 0.05 of each other — the near-even trade between overlap and obliquity becoming exact as the edge effects shrink.