OOptimal Template Banks
Bruce Allen
Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Leibniz Universit¨at Hannover, Callinstrasse 38, D-30167, Hannover, Germany (Dated: February 23, 2021)When searching for new gravitational-wave or electromagnetic sources, the n signal parameters(masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discreteset of points in parameter space. The computational cost is proportional to the number of thesepoints, and if that is fixed, the question arises, where should the points be placed in parameter space?The current literature advocates selecting the set of points (called a “template bank”) whose Wigner-Seitz (also called Voronoi) cells have the smallest covering radius ( ≡ smallest maximal mismatch).Mathematically, such a template bank is said to have “minimum thickness”. Here, we show that atfixed computational cost, for realistic populations of signal sources, the minimum thickness templatebank does not maximize the expected number of detections. Instead, the most detections areobtained for a bank which minimizes a particular functional of the mismatch. For closely spacedtemplates, the most detections are obtained for a template bank which minimizes the average squareddistance from the nearest template, i.e., the average expected mismatch. Mathematically, such atemplate bank is said to be the “optimal quantizer”. We review the optimal quantizers for templatebanks that are built as n -dimensional lattices, and show that even the best of these offer only amarginal advantage over template banks based on the humble cubic lattice. I. INTRODUCTION
Many searches for gravitational-wave and electromag-netic signals are carried out using matched filtering,which compares instrumental data to waveform tem-plates [1–3]. Because the parameters of the sources arenot known a priori, many templates are required, form-ing a grid in parameter space [4–7]. Like the mesh on afishing net, the grid needs to be spaced finely enough thatsignals don’t slip through. But if the grid has far morepoints than are needed, the computational cost becomesexcessive. For this reason, a substantial technology hasevolved to create these grids [8–14].What choice of grid is optimal, for a particular numberof grid points? The literature on the topic answers thequestion as follows: select the grid which minimizes thelargest distance between any point in parameter spaceand the nearest grid point [14].If the grid is an n-dimensional lattice, then this choicecorresponds to picking the lattice of minimum “thick-ness”. That is to say, it selects the lattice whose Wigner-Seitz (WS) cells (also called Voronoi cells, Brillouinzones, and Dirichlet cells) have the smallest maximumradius. (An introduction to lattices and a description ofthe “classical” lattices may be found in Chapters 2 and4 of [15].)Here, we show that the minimum-thickness grid is notthe best choice: it does not minimize the number of sig-nals which are “lost” because of the discreteness of thegrid. For that purpose, and provided that the grid pointsare not too widely separated, the best choice is the gridthat minimizes the (normalized) second moment, whichis the mean value of the squared distance (mismatch) tothe nearest grid point. If the grid is a lattice, then math-ematicians call such a lattice the “optimal quantizer”.In this paper, we obtain simple expressions for the frac- tion of lost sources, in terms of the moments of the grid.If the grid is closely spaced, this expression only involvesthe second moment. For more widely spaced grids, wereplace the usual quadratic approximation for the mis-match with a more accurate spherical approximation [16].The resulting expression for the fraction of sources lostinvolves all of the even moments, although in most casesthe first half-dozen even moments is sufficient for an ac-curate approximation.The main results of this paper apply to any grid oftemplates in parameter space. However, in many appli-cations a regular grid is desirable; this may be system-atically constructed as an n -dimensional lattice [15]. Alattice is obtained from a set of n linearly-independentbasis vectors, forming all linear combinations with inte-ger coefficients. For a lattice, all WS cells are identicalunder lattice translation. In addition to “grid” and “lat-tice” there is a third type of object that arises in thispaper, which we will call a “tessellation”. We define thisas set of points whose WS cells have identical size andshape, but may be oriented differently (via reflection androtation). This is also called a “packing”.The paper is structured as follows. In Section II webriefly review matched filtering, templates and templatebanks, the overlap between templates, and the mismatchfunction m on parameter space. In Section III we showhow, using the mismatch as a distance measure, the pa-rameter space is broken up into WS cells surroundingeach template. The radius of the smallest sphere whichencloses one of these WS cells is called the covering radius(or WS radius) of the grid. In Section IV we review theconventional wisdom for template placement, which isto select the template grid points so as to minimize thecovering radius R for a given average WS cell volume.This minimizes a quantity known as the thickness of thelattice. Section V contains the main results of this pa- a r X i v : . [ a s t r o - ph . I M ] F e b per: we calculate the fraction of detections which are lostbecause of the discreteness of the template bank. Mini-mizing the fraction of lost detections for a fixed numberof templates (i.e., at fixed computing cost) is achieved byminimizing a particular functional of the mismatch, givenin Eq. (5.9). For closely spaced templates, this amountsto minimizing the second moment of the template grid,as shown in Eq. (5.6). In the mathematical literature,grids which minimize this quantity are called “optimalquantizers”. In Section VI we extend these results to thecase where the putative signals are not uniformly dis-tributed in parameter space. Lastly, in Section VII wediscuss possible choices of template grids, and summarizethe current state of knowledge about optimal quantizerswhen the grids are lattices or tessellations. This is fol-lowed by a short conclusion.This paper concentrates on the case of closely-spacedtemplates (small mismatch m ≈ r , valid for r (cid:28) m ≈ sin r , valid for r (cid:46) π/ II. MATCHED FILTERING AND THEOVERLAP BETWEEN TEMPLATES
The classic signal detection problem is the following.We have instrumental or detector data in the form of acontinuous or discretely sampled time series s ( t ), whichmight or might not contain a weak signal with waveform T ( t ) and unknown amplitude α . The signal data streamis contaminated with zero-mean additive noise n ( t ), so s ( t ) = n ( t ) + αT ( t ) . (2.1)The problem is to identify (with some desired confidence)if the weak signal is present, and to estimate its ampli-tude.The classic solution to this problem is called linearmatched filtering [5, 6, 18–25]. This takes as input thedata stream s and the template T and produces as outputa single value, which is a positive-definite inner product ρ = ( Q T , s ) , (2.2)where Q T = T / ( T, T ) / is the “optimal filter” or“matched filter” associated with the template T . Theinner product may be computed in the time or frequencydomain; the optimal choice depends upon the propertiesof the noise. For example, if the noise is stationary andGaussian, then the inner product is( A, B ) = (cid:90) A ∗ ( f ) B ( f ) N ( f ) df, (2.3)where on the right-hand side the functions A, B havebeen transformed to the frequency domain, and N ( f ) isthe power spectrum of the noise n ( t ). This inner productsuppresses frequencies where the noise is large. The expected value of ρ (with a fixed signal and manyinstances of noise) is α ( T, T ) / , and the expected valueof ρ is α ( T, T )+1. The square of ρ is called a “detectionstatistic”; large values indicate that the signal is likelypresent. Since the variance of ρ is unity, the actual orexpected values of ρ , | ρ | , and/or ρ are called the signal-to-noise ratios (SNR). For Gaussian noise, the statisticalsignificance (log of the likelihood ratio) is proportionalto ρ . This is reviewed in a signal-processing context in[26, 27] and in the gravitational-wave (GW) context in[28] and [29].If there was only a single possible signal waveform,then one template T and one filter Q T would suffice.However, in most cases of interest, the signal waveformis dependent upon a number of unknown parameters. Forexample, the gravitational-wave signals produced by theinspiral of two masses depend upon the values of themasses, the sky location of the system, the spins of thetwo bodies, and the relative orientation and shape of thebinary orbit. Here n denotes the dimension of that pa-rameter space and λ a with a = 1 , · · · , n are coordinateson that space. We use λ with no superscript to refer tothe collection of these coordinate values.Since the signal parameters are unknown, the templatecannot match them precisely. This reduces the SNR com-pared to a perfect-match template, which has expectedSNR ρ max = α ( T, T ) / . The mismatch m is easy tocharacterize. In a mismatched template T (cid:48) , with corre-sponding filter Q T (cid:48) , the expected (detected) SNR wouldbe ρ det = α ( T (cid:48) , T ) / ( T (cid:48) , T (cid:48) ) / . The mismatch m is thefractional loss m = ρ − ρ ρ = 1 − ( T (cid:48) , T ) ( T, T )( T (cid:48) , T (cid:48) )= 1 − cos ( θ ) = sin ( θ ) , (2.4)where θ is the angle between T and T (cid:48) . It follows imme-diately that m lies in the unit interval m ∈ [0 , T i must be employed,where i labels the template, corresponding to a signalwith parameters λ ai . A set of i = 1 , · · · , M discrete points λ ai in parameter space is called a template bank with M templates. Each template has an associated matched fil-ter Q i = Q T i A search of the instrumental output data s is carriedout as follows. For each template in the bank, the SNR ρ i = ( Q i , s ) is computed. If any of the ρ i are abovethe detection threshold ρ D , a detection is claimed [30].In most cases, the parameters of the source are close tothose of the template which registered the largest SNR.While the signal parameters λ a might be close to theparameters λ ai of one (or more) of the templates T i , theywill never be precisely the same; ρ will be decreased bythe parameter mismatch. If this causes ρ to dip be-low the detection threshold in all templates, a potentialdetection would be missed. To quantify this, one usesEq. (2.4) to define a mismatch function m ( λ ) ≥ m ( λ ) = min i =1 , ··· ,M (cid:20) − ( T λ , T i ) ( T λ , T λ )( T i , T i ) (cid:21) , (2.5)where T is placed at λ a and T i = T ( λ ai ) ranges over allof the different templates in the bank.By definition, m ( λ ) vanishes at the locations of thetemplates λ = λ i . Because m is non-negative andsmooth, it has quadratic behavior near these minima.Thus, for λ a near the parameters λ ai of the i ’th template,one has m ( λ ) ≈ r = g ab ∆ λ a ∆ λ b , (2.6)where ∆ λ a = λ a − λ ai , and g ab is a positive-definite sym-metric quadratic form called the parameter-space metric.In general, g ab will depend upon λ , but to simplify thetreatment that follows, we will assume that the metric isindependent of the coordinates and thus flat. Then, thequantity r is precisely the squared distance to the near-est point in the template bank. An example is shown inFigure 1.It follows from Eq. (2.5) that r and m are dimensionlessquantities, so while we often refer to r as a “distance”, itis not a physical length. For this reason, in this paper,quantities which are independent of an overall re-scalingof r are called “scale invariant” rather than “dimension-less”.Most of the literature assumes the quadratic approxi-mation for the mismatch given in Eq. (2.6). While thisis valid provided that m <<
1, it is unbounded above,whereas by definition the mismatch is bounded above by m ≤
1. Recent work [16] has shown that in many cases abetter approximation to the mismatch is the “sphericalapproximation” m ( λ ) = sin ( r ). This approximation isalso bounded to the correct range.In what follows, we will investigate both approxima-tions to the mismatch, and their consequences. For clar-ity, we will use r = g ab ∆ λ a ∆ λ b to denote the quadraticapproximation to the mismatch, and sin r to denote thespherical approximation. III. WIGNER-SEITZ CELLS
Using the mismatch m as a “distance measure”, theparameter space for a given template bank may be par-titioned into WS cells [31] which are in one-to-one cor-respondence with the templates: there is one cell sur-rounding each template. The WS cells are defined bythe property that the points in a given cell have smallermismatch to that cell’s template than to any other tem-plate.These cells were also studied by Dirichlet [32], Voronoi[33–36] and Brillouin [37]. Since Voronoi was the firstto investigate them in arbitrary dimension for arbitrarygrids, it would be fair to use his name for them. How-ever, in this paper we use “V” to denote volume, whereas“WS” is unambiguous. FIG. 1. The plane represents a two-dimensional parameterspace with coordinates λ , λ . Templates have been placedon a lattice formed from the vertices of equilateral triangles.The vertical axis shows the squared-distance r ( λ , λ ), tothe nearest template point. The zeros of the function are atthe template locations, which here form an A ∗ lattice. Thediscontinuities in the gradient of r lie on the boundaries ofthe Wigner-Seitz (WS) cells, which are hexagons. The WSradius R is the maximum value of r . Any given template bank has a maximum value of r ,which here we denote by R and call the “Wigner-Seitzradius”. An example can be seen in Figure 1. R is alsocalled the “covering radius”: it is the radius of the small-est ball [38] which encloses all points of the WS cell.Traditionally, template banks have been constructedby (a) deciding how many templates are needed, then (b)placing the grid points so as to obtain the smallest pos-sible value of R . This “traditional wisdom” correspondsto minimizing the thickness of the template grid. IV. THICKNESS, AND TRADITIONALTEMPLATE PLACEMENT
The thickness (also called “covering density”) Θ ≥ n -dimensional lattice. Suppose that the lattice includes theorigin, let WS denote the Wigner-Seitz cell of the origin,and let V ( W S ) denote its n -volume. Let R denote theWS “covering radius”: the maximum distance r betweenthe origin and any point in WS. Then the thickness isdefined as Θ = V ( B n ( R )) V ( W S ) , (4.1)where B n ( R ) is an n -ball of radius R , and V ( B n ( R )) = π n/ R n / Γ( n/ n -volume. Since the ball coversthe WS cell, its volume cannot be smaller than that ofthe WS cell. Thus, by definition, Θ ≥ f , then R isre-scaled by the same factor, and the volumes V ( B n ( R ))and V ( W S ) are both re-scaled by f n , leaving Θ invariant.The thickness is a measure of the way in which theballs of radius R “overcover” space. If a ball of radius R is placed around each lattice point, then Θ is the averagenumber of balls in which a random point of parameterspace lies, where we assume that the parameter spaceis large enough to contain many lattice points. A com-prehensive review of classical lattices and their thicknessand other properties may be found in Chapters 2 and 4of [15].The existing literature on template placement assertsthat the best choice of grid is the one which has min-imal thickness Θ. The idea is that the available com-puting capacity determines the number M of templateswhich can be employed, which means that V ( W S ) isfixed, since M · V ( W S ) is the total volume of parameterspace. Picking the grid with the smallest covering radiusthen ensures that the worst case mismatch (in either thequadratic or the spherical approximation) is minimized.This in turn minimizes the thickness, Eq. (4.1). For ex-ample [14] states, “
The construction of optimal templatebanks for matched-filtering searches is an example of thesphere covering problem.... An optimal template banktherefore consists of the smallest possible number of tem-plates that still guarantees that the worst-case mismatchdoes not exceed a given limit. ”. A recent textbook [28],Section 7.2.1, says, “
The problem of constructing a gridin parameter space is equivalent to the so-called coveringproblem.... The optimal covering would have minimumpossible thickness. ”We will shortly show that, under reasonable assump-tions, this choice, i.e. minimizing the thickness, is not optimal. If the goal is to maximize the expected num-ber of detections for a given number of templates (i.e.,at fixed computing power), it is better to place the tem-plates to achieve the smallest average value of r (if r issmall) or on some other combinations of moments (if r is not small).This fallacy, that the minimum thickness templatebank is the best choice, may have a historical basis.The search for gravitational waves was the initial mo-tivation for the study of matched-filter template banksin the 1990s, and for the development of the technologyfor template bank placement in the two decades that fol-lowed. Until the first signals were detected in late 2015[39] the community focused on obtaining the most con-straining “upper limits”. These are upper bounds (with astated statistical confidence) on the strength of differentpossible gravitational-wave sources: if a stronger sourcehad been present, it would have been detected. For thatpurpose, obtaining the most constraining upper limits,which apply strictly over the entire parameter space, aminimum thickness template bank is optimal.This has some important implications. Let’s com-pare the cubic lattice Z n to the A ∗ n lattice, which isan n − dimensional generalization of the 2-dimensionalhexagonal lattice. For dimensions n ≤
17, as explainedin [14], A ∗ n is either the thinnest or close to the thinnestclassical root lattice. (Thinner non-classical lattices have been constructed numerically, see Table 2 in [40].)The thickness of Z n is easily computed from Eq. (4.1),giving Θ[ Z n ] = ( πn/ n/ Γ( n + 1) , (4.2)whereas the thickness of the A ∗ n lattice is [15, 17]Θ[ A ∗ n ] = ( πn ( n + 2) / n/ ( n + 1) (1 − n ) / Γ( n + 1) . (4.3)As the parameter-space dimension n → ∞ , the ratioasymptotes to Θ[ Z n ] / Θ[ A ∗ n ] → n/ / √ n + 1, and Z n becomes much thicker than A ∗ n . This is illustrated inFigure 2 of [14]. For example, in 8 dimensions, Θ[ Z ] =2 π / ≈ .
94, whereas Θ[ A ∗ ] = 20000 π / ≈ . A ∗ lattice.However, we will see shortly that when ranked by lostdetections, these two lattices are quite similar. V. DETECTIONS LOST FROM TEMPLATEMISMATCH
We now examine how the choice of template locationsinfluences the expected number of detections.Assume that the sources populate the parameter spaceuniformly, and have a distribution of expected SNR val-ues (in perfectly matched templates) which is describedby a population distribution function P , so that dN = P ( ρ ) dρ is the number of sources in the SNR range( ρ , ρ + dρ ). If the signal amplitude is inversely pro-portional to distance (cid:96) (as is the case for GWs), then ρ ∝ /(cid:96) . For sources uniformly distributed in a flatd=2-dimensional Galactic disk, one therefore has dN ∝ (cid:96)d(cid:96) ∝ ρ − dρ , and for sources uniformly distributed inflat d=3-dimensional space, one has dN ∝ (cid:96) d(cid:96) ∝ ρ − dρ [41]. Thus we take dN = P ( ρ ) dρ = d N D (cid:18) ρ ρ (cid:19) d/ dρ ρ , (5.1)where d is the effective dimension [42] of the source distri-bution and N D is the total number of detectable sources(i.e., sources with SNR above the detection threshold ρ D ).Suppose that the parameter space is densely coveredwith a very large number of closely-spaced templates. Inthis case, the expected number of detections is N D = (cid:90) dN = (cid:90) ∞ ρ D P ( ρ ) dρ . (5.2)This is the best-case scenario.Now consider the more realistic case, where the tem-plates are spaced at mismatch m . The expected numberof detections is reduced, because some of the population,whose SNR would be only slightly above the threshold ifthere was a perfectly matching template, will fall belowthe detection threshold, due to mismatch losses to thenearest template. To be detectable the SNR must satisfy ρ (1 − m ) ≥ ρ . So the expected number of detections is N found = V − (cid:90) (cid:90) ∞ ρ D / (1 − m ( λ )) P ( ρ ) dρ dV. (5.3)Here, m ( λ ) denotes the mismatch to the nearest tem-plate, the outer integral is over the n-dimensional pa-rameter space with invariant volume measure dV = √ det g d n λ , and V = (cid:82) dV is the n -volume of parame-ter space.The number of detections “lost” because of the finitespacing of the template bank is the difference N lost = N D − N found , which is therefore N lost = V − (cid:90) (cid:90) ρ D / (1 − m ( λ )) ρ D P ( ρ ) dρ dV. (5.4)We now investigate several limits of this expression.A simple limit is obtained if the template bank isclosely spaced, so that everywhere in the parameter spacethe maximum mismatch m is small compared to unity,implying that m ≈ r . In this limit ρ D / (1 − m ) ≈ ρ D (1 + r ), so that N lost ≈ V − (cid:90) (cid:90) ρ D + ρ D r ( λ ) ρ D P ( ρ ) dρ dV ≈ V − ρ D P ( ρ D ) (cid:90) r ( λ ) dV. (5.5)Making use of Eq. (5.1), the fraction of lost detections isthen N lost N D = d V − (cid:90) r ( λ ) dV = d (cid:104) r (cid:105) , (5.6)where the final equality serves to define the “average sec-ond moment” (cid:104) r (cid:105) of the template grid.Thus, the number of “lost” detections is determinedby the average value of the mismatch over the templatebank. If the template bank is a lattice or tessellationin parameter space, then the fraction of lost detections(compared with a very closely-spaced template bank) is N lost N D = d V − W S (cid:90)
W S r dV, (5.7)where the integral is over a single WS cell of volume V W S ,and we have assumed that the parameter space containsa large number of such cells.The computational cost is determined by the numberof templates at which the SNR is calculated, so fixingthe computational cost is equivalent to fixing the numberof templates, or fixing the volume V W S . Thus, at fixedcomputational cost, if the templates are closely spaced,the number of lost signals is minimized by minimizingthe average mismatch over the template bank . In the mathematical literature, the quantity (cid:82)
W S r dV /V W S is called the “normalized secondmoment of the lattice”.
The lattice that minimizes thisquantity in n -dimensions, for fixed Wigner-Seitz cellvolume V ( W S ) , is called the optimal quantizer [15]. To compare lattices, it is conventional to introduce ascale-invariant quantity G , defined in Eq. (7.1). In termsof this quantity, for a closely-spaced template bank, N lost N D = n d V W S ) /n G [lattice] . (5.8)Thus, the relative number of lost signals at fixed comput-ing cost for two closely-spaced lattices can be estimatedfrom the ratio of the lattice’s scale-invariant quantizationconstants G .If the mismatch m is not small, then the fraction oflost signals also depends upon the higher moments of thegrid. One example is the search for continuous gravita-tional waves (CW) from rapidly spinning neutron starsin the Galactic disk, which have approximately a d =2 di-mensional distribution. The parameter space for an un-informed search is very large, and so these searches arecomputationally limited, and often carried out at largemismatch. Let us assume a d -dimensional source distri-bution, and use Eq. (5.1) to evaluate the inner integralin Eq. (5.4). The fraction of lost detections is then N lost N D = V − (cid:90) (cid:16) − (1 − m ( λ )) d/ (cid:17) dV ≈ V − (cid:90) (cid:0) − cos d r (cid:1) dV. (5.9)In the final line, we have used the spherical approxima-tion [16] to the mismatch m = sin r . This approxima-tion only holds in the interval r ∈ [0 , π/ r > π/ m = 1.)If the grid is a lattice, then the integral can be replacedby the integral over a single WS cell. If the WS radius R < π/ N lost N D = d (cid:104) r (cid:105) − d (3 d − (cid:104) r (cid:105) + d (15 d − d + 16)720 (cid:104) r (cid:105)− d (cid:0) d − d + 588 d − (cid:1) (cid:104) r (cid:105) + · · · , (5.10)where < r p > = V − W S (cid:90)
W S r p dV (5.11)denotes the normalized p ’th moment of the WS cell.Several authors have investigated how continuousgravitational-wave searches should be structured, to pro-vide maximum sensitivity at fixed computing cost [43–46]. Their results, which assume closely-spaced tem-plates, foreshadow ours. While those papers and laterwork do not explicitly discuss the optimization of a tem-plate bank or lattice, they indicate that the optimalsearch-parameter choices (for example, stack-slide timebaseline) and achievable population-averaged sensitivity[46–49] are determined by the average value that themismatch takes over the template bank, and not by thebank’s thickness. VI. NON-UNIFORM POPULATION DENSITYAND THRESHOLD
Earlier in this Section we assumed that the sourcespopulate the parameter space uniformly, and that thedetection threshold is independent of the source type.Both of these assumptions can be dropped. Let dN = P ( ρ , λ ) dρ dV be the expected number of sources inthe parameter-space volume dV with SNR in the range( ρ , ρ + dρ ). Then the more general result is N lost = V − (cid:90) (cid:90) ρ D ( λ ) / (1 − m ( λ )) ρ D ( λ ) P ( ρ , λ ) dρ dV. (6.1)The small-mismatch limit is easily obtained, showingthat the number of lost signals is determined by the av-erage value of the mismatch, appropriately weighted bythe number of sources in that region of parameter space. VII. CHOICE OF OPTIMAL LATTICE
In a given parameter-space dimension n , what latticeis optimal? In contrast, how much is lost if a non-optimallattice is selected? How would this compare with a lat-tice selected for minimum thickness (smallest coveringradius)?If the mismatch is large (i.e., the quadratic approxima-tion cannot be used), then these questions are not easilyanswered. In a companion paper [17] we have computedthe fraction of lost detections for the cubic lattice Z n andthe A ∗ n lattice, which is an n -dimensional generalizationof the hexagonal lattice.If the mismatch is small enough that the quadraticapproximation m ≈ r is valid, then we have shown inEqs. (5.6) and (5.7 that the optimal lattice is the onethat minimizes the normalized second moment of the lat-tice. The n -dimensional lattice that minimizes this sec-ond moment (for fixed V ( W S )) is known as the “optimal n -dimensional quantizer”. To easily compare the secondmoment of n -dimensional lattices at fixed V ( W S ), it isconventional to introduce the scale-invariant “quantiza-tion constant” G = 1 n (cid:82) W S r dV (cid:0)(cid:82) W S dV (cid:1) n . (7.1)In contrast to the normalized second moment (cid:104) m (cid:105) , thequantization constant G has the property that its valueis invariant under uniform re-scaling of the lattice. Thefactor of n − appearing in the definition ensures that the cubic lattice Z n has a scale-invariant second moment G =1 /
12 = 0 . · · · which is independent of dimension.Table I summarizes the current state of knowledge fordimensions n <
16: it shows the lattices which have thesmallest-known quantization constants G , along with ref-erences. In dimensions 7 and 9, the best currently knownquantizers are the non-lattice tessellations D +7 and D +9 ;for completeness we have also listed the best currentlyknown lattice. For comparison, we have also listed the“classical” root lattices with the smallest known thick-ness. (For n ≥
6, thinner lattices have been constructednumerically, by semidefinite optimization in the space oflattices; see Table 2 of [40]).What is remarkable, and immediately visible from Ta-ble I, is that for small mismatch, where the quadraticapproximation m = r applies, the best lattices, withtypical G ≈ .
07, have only a very marginal advantagein terms of lost signals when compared with the humblecubic lattice Z n , with G = 1 / ≈ . B n provides a lowerlimit for the scale-invariant second moment G . One ob-tains G [Any grid] ≥ G [ B n ] = Γ( n/ /n π ( n + 2) . (7.2)One can evaluate G [ B n ] in the n → ∞ limit using Stir-ling’s approximation, showing that G > / π e ≈ . Z n , for closely spaced templates the best choice of gridcan at most reduce the fraction of lost signals by a factorof N lost [ Z n ] N lost [Best n -grid] ≤ G [ Z n ] G [ B n ] < π e6 ≈ . . (7.3)In practice, the factor is substantially smaller than this.For example, in 4 dimensions the best known quantizeris D , which in comparison with the Z lattice wouldreduce the fraction of lost detections by about 9%, since G [ Z ] /G [ D ] ≈ . E , whose fractional advantage over the Z lattice is about 16%, since G [ Z ] /G [ E ] ≈ . Z n lattice with closely-spaced templates, thefraction of lost signals in Eq. (5.7) takes very simple form.Since G = 1 /
12, we have N lost N D = n d
24 ( V W S ) /n = d R = d m worst , (7.4)where R is the WS radius and m worst is the worst-casemismatch in the quadratic approximation. Thus, for a d = 3-dimensional distribution, at a worst-case mismatchof 20%, about 10% of signals would be lost. TABLE I. The smallest quantizer constant (smallest G ) lat-tices currently known for low dimension n . Also listed arethe thinnest (smallest Θ) classical root lattices (but note thatthinner non-classical lattices have been constructed for n ≥ G yields the fewest “lost”detections (for small mismatch). In dimensions 7 and 9, thebest known quantizers are the non-lattice tessellations D +7 and D +9 , see text, footnotes, and [50] for details. dimension Lattice Thickness Second Ball limit n Θ Moment G on G a A ∗ = Z b c A ∗ b c A ∗ b c D d c A ∗ b c D ∗ d c A ∗ b c E ∗ e n A ∗ b f D +7 m g E ∗ h o A ∗ b i E d c A ∗ b i D +9 m g AE p j A ∗ b i D +10 m k A ∗ b i A ∗ b i K d l A ∗ b i A ∗ b i A ∗ b i A ∗ b i a Eq. (7.2). b Eq. (4.3) or Table 2.1 in Conway and Sloane [15]. c Table 2.3 in [15]. d Table 2.1 in [15]. e Use R and det following Ch. 4 Eq. (126) in [15]. f Ch. 21 Eq. (51) in [15]. g Non-lattice packing (tessellation). See Agrell and Eriksson [50]and Notes on Ch. 2 in [15]. Exact values were found by Sikiri´c[51] using the methods of [52] and are G [ D +7 ] = 178751 / G [ D +9 ] = 924756607 / h Use R and det following Ch. 4 Eq. (115) in [15]. i Appendix of Allen and Shoom [17]. j Lattice of Eq. (31) of [50], not a classical lattice, denoted herewith the initials of Agrell and Eriksson. k Identified in [50]; exact value from Sikiri´c et al. [52]. l Exact value from [52]. m Text before Ch. 4 Eq. (94) of [15] with last paragraph [50]Section 3. n Estimated for Table 2.3 of [15]; exact value from Worley [53]. o Estimated for Table 2.3 of [15]; exact value from Worley [54]. p E. Agrell, private communication. The deep holes of the lattice B of Eq. (31) of [50] have the form ( ± , , , , , , , , ± a ),where a ≈ . R = (1 + a ) / . Thevolume of the WS cell is det( B ) = 2 a , givingΘ = π / R / / det( B )Γ(11 /
2) = 16 π (1 + a ) / / a . VIII. CONCLUSION
In this paper, we have shown in Eq. (5.9) how to quan-tify the fraction of detections which are lost because ofthe discreteness of a template bank; these sources couldhave been detected had the templates been more finelyspaced. The fraction depends upon the properties of thesource distribution and upon the placement of the tem-plates. If the templates are not too far apart, the latterdependence is through the average value of the mismatch(second moment of the distance) as in Eq. (5.7).For simplicity, our source models Eq. (5.2) assumetime-independent source distributions with “Euclidean”volume measures. This is sufficient if sources are notat cosmological distances, so that the large-scale geom-etry of space-time does not influence the measure, andif the sources are closer than cτ , where τ is the timescale on which the properties of the source distributionevolve, and c is the speed of light. Future generations ofgravitational-wave detectors will have a reach which ex-tends to the Hubble radius, and will study sources whichhave significant evolution over redshifts of a few. Forthose, a precise estimate of “lost” sources may requirepopulation models that incorporate source and/or cos-mological evolution.To maximize the expected number of signal detec-tions for a given number of templates, we have shownthat a template bank must minimize the average of afunction of the mismatch m , given in Eq. (5.9). Forclosely spaced templates, where the mismatch reducesto the squared distance to the nearest template, this cor-responds to choosing a grid which is the “optimal quan-tizer” as in Eq. (5.8), which minimizes the average valueof the squared distance to the closest point in the tem-plate bank. This contrasts with standard wisdom, whichholds that the optimal choice of template bank is the onewhich minimizes the covering radius (or equivalently, thethickness).Template bank thickness is relevant for upper limits,but it is necessary to distinguish between two types ofupper limits: “strict” and “population-averaged”. Strictupper limits apply at every point in parameter space,whereas the population-averaged upper limits only applyon average (with the stated confidence) to the entire pop-ulation. The literature contains examples of both. Some-times (for example see [56–59]) both variants are givenin the same paper. While the thinnest template bankwill give the most constraining strict upper limit, it doesnot maximize the expected number of detections, and isprobably also not optimal for the population-averagedupper limits.Often, template banks are constructed as regular lat-tices. To compare two lattices in the closely spaced case,and to identify which choice maximizes the expectednumber of detections for a fixed number of templates,one need only compare the scale-invariant second mo-ment (quantization constant) G of the lattice. The ra-tios of G for two lattices is proportional to the relativenumbers of “lost” detections at fixed computing cost, ascan be seen from Eq. (5.6).This has an important consequence for the humble cu-bic lattice Z n . While it has much thinner and more so-phisticated cousins such as A ∗ n , the ratios of their quan-tization constants G are not far from unity. This can beseen from Table I, keeping in mind that for the cubic lat-tice, G = 1 / ≈ . Z n lattice is very thick, because the corners of thecube “stick out”, giving it a large covering radius. Thismakes it a poor choice for obtaining strict upper limits,because a signal hidden in one of those distant corners atradius R could have much larger amplitude than the bulkof the population, yet might still go undetected. How-ever, if the goal is detection (or a population-averagedupper limit), this does not matter. The volume in thecorners is quite small, which in turn means that the ex-pected number of signals lost there is also small [60].There are many types of computationally limited sig-nal searches, for which these results are relevant. Forexample, it is currently not possible to do an all-skysearch for gamma-ray pulsations in binary systems, orfor continuous gravitational waves from neutron stars inbinary systems. The parameter space here (counting di-mensions in parentheses) includes sky-position (2) andfrequency and spindown (2). For circular orbits one ad-ditionally has orbital period, inclination angle, and mod-ulation depth (3); if the orbit is eccentric, then two ad- ditional parameters are needed. So in this case, the pa-rameter space is 7- or 9-dimensional [61]. The situationis even worse for gravitational-wave searches from binaryinspiral systems where spin effects are significant; for cir-cular orbit systems there are 14 parameters [6]. Currenttechnology does not have the computational power toexplore such large dimensional spaces, but advances inquantum computing may permit such searches in the fu-ture.A companion publication [17] looks in more detail atthe case where the templates are not closely spaced, mak-ing use of the spherical approximation [16] to the mis-match, m = sin r . IX. ACKNOWLEDGMENTS
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