Template banks based on \mathbb{Z}^n and A_n^* lattices
TTemplate banks based on Z n and A ∗ n lattices Bruce Allen ∗ and Andrey A. Shoom † Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Leibniz Universit¨at Hannover, Callinstr. 38, D-30167, Hannover, Germany (Dated: February 24, 2021)Matched filtering is a traditional method used to search a data stream for signals. If the source(and hence its n parameters) are unknown, many filters must be employed. These form a grid inthe n -dimensional parameter space, known as a template bank. It is often convenient to constructthese grids as a lattice. Here, we examine some of the properties of these template banks for Z n and A ∗ n lattices. In particular, we focus on the distribution of the mismatch function, both in thetraditional quadratic approximation and in the recently-proposed spherical approximation. Thefraction of signals which are lost is determined by the even moments of this distribution, which wecalculate. Many of these quantities we examine have a simple and well-defined n → ∞ limit, whichoften gives an accurate estimate even for small n . Our main conclusions are the following: (i) a fairlyeffective template-based search can be constructed at mismatch values that are shockingly high inthe quadratic approximation; (ii) the minor advantage offered by an A ∗ n template bank (comparedto Z n ) at small template separation becomes even less significant at large mismatch. So there islittle motivation for using template banks based on the A ∗ n lattice. I. INTRODUCTION
Matched filtering is a standard technique [1, 2] used tosearch for weak gravitational-wave signals from the bi-nary inspiral of black holes and/or neutron stars. Thiscompares the data (suitably weighted in frequency space)to a template of the expected waveform [3–14]. Matchedfiltering is also used to search for weak electromagnetic(radio and gamma-ray) [15] and gravitational-wave sig-nals from rapidly rotating neutron stars (pulsars) [16]and has many other applications across a broad range offields and topics.Because these searches are typically looking for newevents and/or unknown sources, the parameters of thesignals are not known. Some examples of these param-eters include sky position, mass, and spin or chirp fre-quency. Thus, a collection of templates must be em-ployed. The grid of these templates in parameter spaceis generally referred to as a “template bank”.If the parameter space is low-dimensional and theparameter-space volume is not too large, one can sim-ply “overcover” the space, putting many redundant tem-plates close together. However, if the parameter-spacedimension and/or volume is large, this quickly becomes(computationally speaking) very expensive. On the otherhand, if the templates are spaced too far apart, thenit’s possible that some signals be missed, because therewas no template in the bank which matched the wave-form well enough. Thus, a compromise must be reached:enough templates must be employed that signals are notlost, but the number of templates must not be so largethat the computing cost explodes. For some searches(e.g., for continuous gravitational waves from neutron ∗ [email protected] † [email protected] stars in binary systems) the computing cost is so highthat it constrains the search sensitivity.The problem of how to place templates in parameterspace is well studied. There are many ways to constructtemplate banks. For example, one can simply place thetemplates at random [17], with a high enough densitythat most signals are likely to lie near enough to a tem-plate. Or one can improve this by removing redundanttemplates which are “too close” to neighboring ones, andadding more templates at random, if required [18]. Al-ternatively, one can build a template bank as a regularlattice in parameter space. Two examples of such lat-tices are the Z n and A ∗ n lattices. The first of these isjust the Cartesian product of equally spaced grids in alldimensions, and the second is the n -dimensional general-ization of the two-dimensional hexagonal lattice and thethree-dimensional face-centered cubic (fcc) lattice.One way to characterize a template bank is via the mis-match function m . This is a function on parameter space,which quantifies how much signal-to-noise ratio (SNR) islost because of the discreteness of the template bank. Itsvalue at any point is the fractional difference between thesquared SNR obtained for a signal with those parameters,and the squared SNR that would have been obtained hada template been located at that point. Thus, m vanishesat the locations of the templates, and is largest “halfwayin-between” two templates. In a recent paper, we showhow the fraction of lost signals is related to the averageof m and functions of m [19].When the templates are close together, and the mis-match is small, m can be expressed as a positive-definitequadratic form and thought of as the squared distancebetween the parameter-space point and the closest tem-plate. Thus, in this approximation, m ≈ g ab ∆ λ a ∆ λ b ,where g ab is the metric on the parameter space and∆ λ a is the coordinate separation between the two points(see, e.g., [13, 20]). Here, we call this the quadratic ap-proximation to the mismatch, and write it as m = r . a r X i v : . [ a s t r o - ph . I M ] F e b When the templates are less-closely spaced, a better ap-proximation to the mismatch is the “spherical” ansatz, m ≈ sin r = sin ( (cid:112) g ab ∆ λ a ∆ λ b ), recently introduced in[21].If the mismatch is small, then the bank which mini-mizes the average second-moment of r loses the small-est fraction of signals [19]. If the bank is a lattice, thisis called the “optimal quantizer” [22]. This paper ex-tends those results to large mismatch, by exploiting thespherical ansatz [21], and carrying out an explicit calcu-lation for template banks constructed from the Z n and A ∗ n lattices.Our paper is organized as follows. In Sec. II we de-scribe the n -dimensional lattices Z n and A ∗ n , and derivetheir key properties. In Sec. III we calculate the fractionof lost detections in the quadratic and spherical approxi-mations for these lattices for 2- and 3-dimensional sourcedistribution. This fraction of lost signals may be thoughtof as the “inefficiency” or “loss fraction” of the lattice.In Sec. IV we evaluate the loss fraction as the parameterspace dimension n → ∞ . This gives simple analytic ex-pressions; in some cases the approach is fast enough thatthese are good approximations even in finite numbers ofdimensions. In Sec. V we compare the loss fraction of Z n and A ∗ n at fixed computing cost. Finally, in Sec. VI, weexamine the distribution function of the squared radius r , and its properties for the Z n and A ∗ n lattices. This isfollowed by a short Conclusion.Our results only depend on the even-order momentsof the Wigner-Seitz (WS) cells of the lattices, which wedenote by (cid:104) r m (cid:105) . Appendix A contains a calculation ofthese moments for the Z n lattice, and Appendix B con-tains the corresponding calculation for the A ∗ n lattice. II. THE Z n AND A ∗ n LATTICES
The lattices Z n and the A ∗ n are an infinite collection ofregularly-spaced points in Cartesian space R n . We use x to denote a point in R n with the normal Euclidean norm | x | = x · x , where the dot denotes the standard dot prod-uct. The lattices are generated by a set of n basis vectors e i ∈ R n , for i ∈ , · · · , n , which for these particular lat-tices are normalized so that e i · e i = 1. Two-dimensionalrepresentatives of these lattices are illustrated in Fig. 1.To characterize the geometry of the lattice, we shalluse x i to denote Cartesian coordinates and y i to denotelattice coordinates. Accordingly, the lattice consists of allpoints x = y i e i in R n , such that y i = c i (cid:96) , where c i ∈ Z are integers and (cid:96) is the lattice spacing. These points arecalled lattice vertices . From here forward, we shall usethe “summation convention” that repeated indices aresummed.The squared distance r between points x A and x B with lattice coordinates y Ai and y Bi is then r = ( x A − x B ) · ( x A − x B ) = ∆ y i ∆ y j e i · e j = g ij ∆ y i ∆ y j , (2.1) where ∆ y i = y Ai − y Bi are the lattice coordinate separa-tions and the (flat) metric is g ij = e i · e j .The region of R n for which the coordinates y i ’s are suchthat | y i | ∈ [0 , (cid:96) ], is called a “Fundamental Polytope” orFP. The FP has 2 n vertices, which are neighboring latticepoints. The region of R n which is closer [in the sense ofthe coordinate distance Eq. (2.1)] to a given lattice pointthan to any other lattice point is called the “Wigner-Seitz cell” (WS) of that lattice point. We also denote theWigner-Seitz cell of the origin y i = 0 by WS (see Fig. 1).The distance from the origin to the most distant point ofthe WS is called the covering radius or WS radius R ; it isthe radius of the smallest sphere about the origin whichencloses every point of the WS.We can compute the n -volume of the FP and the WSas follows. Since all FP are equivalent, we concentrate onthe fundamental FP defined by lattice coordinate values y i ∈ [0 , (cid:96) ]. The volume of the FP is V FP = (cid:90) (cid:96) dy · · · (cid:90) (cid:96) dy n √ g , (2.2)where g = det( g ij ). The n -volume of the WS, V WS , isequal to that of the FP, because (if the WS is copiedaround all lattice points) they overlap only on the bound-aries (a set of measure zero), are in one-to-one correspon-dence, and cover all of space.The fundamental FP is contractible to the origin, inthe sense that if a point x ∈ R n lies inside it, then sodoes the point λ x for λ ∈ [0 , n − n − R of the WS centeredat the origin by considering the subset of those planeswhich lie in the FP, i.e. which lie halfway between theorigin and the remaining 2 n − n -dimensional ball enclosedby one of the spheres, to the volume V FP = V WS [22],Θ = V ( B n ( R )) V WS . (2.3)Here, V ( B n ( R )) = π n/ R n / Γ(1 + n/
2) is the volume ofan n -ball B n of radius R . From the definition it followsthat Θ ≥
1. Smaller values of Θ indicate less overlapamong the balls, i.e. a more efficient covering. Another quantity used in the literature is the normalized thick-ness (or center density) θ = R n /V WS . ( a ) ( b )FIG. 1. Two-dimensional lattices ( (cid:96) = 1): ( a ) the Z square lattice, ( b ) the A ∗ hexagonal lattice. The fundamental polytropesare shown in light grey, the WS cells are shown in dark grey and inscribed by the dashed circles of the covering radius. Forgeneral n , the basis vectors e i for A ∗ n define vertices of an equilateral n -simplex (see text after Eq. (2.10)). In the following subsections we compute the quantitiesdefined above for the Z n and A ∗ n lattices. We will usethese quantities in calculating the statistical propertiesof functions of the distance, such as the mismatch, forboth the lattices and to compare the derived results. A. The Z n lattice The Z n lattice (see, e.g., [22]) is generated by orthonor-mal basis vectors e i · e j = δ ij , (2.4)where δ ij is the Kronecker delta, i.e., the metric g ij isthe identity matrix. Thus, if the basis vectors are takenas the standard coordinate basis, then the lattice coor-dinates y i are just the normal Cartesian coordinates and x i = y i . The distance function Eq. (2.1) is r = n (cid:88) i =1 ∆ y i , (2.5)and, according to Eq. (2.2), the n -volume of the FP is V FP = (cid:96) n . (2.6)According to Eq. (2.1), the largest distance between anypair of vertices in the FP is r = n(cid:96) . It is also the largestdistance from the origin to a point within the FP.To find the boundary of the WS centered at the ori-gin, we begin by finding the equations of the planes thatlie halfway between the origin and the nearest latticepoints at distance (cid:96) from the origin. (The other potentialbounding planes are irrelevant because they lie outside.)There are 2 n of these nearest lattice points. They havecoordinates (0 , · · · , , ± (cid:96), , · · · , (cid:96) is located inthe j th position and the remaining n − n − n (cid:88) i =1 y i = ( y j ∓ (cid:96) ) + n (cid:88) i =1( i (cid:54) = j ) y i = (cid:96) ∓ (cid:96)y j + n (cid:88) i =1 y i . Thus the planes bounding the WS satisfy y j = ± (cid:96)/ . (2.7)There are 2 n such planes, since j = 1 , · · · , n . Thesedefine an n -cube which is identical to the FP but shiftedby − (cid:96)/ R is easily computed. The point ofmutual intersection of the n planes with y j > n vertices of the WS(defined by intersecting each of the possible planes, onefor each coordinate, n in total) are at the same distance R from the origin. Hence, the WS covering radius R isthe distance of that WS vertex from the origin. Usingthe expression Eq. (2.5) gives R = 14 n(cid:96) (2.8)for the covering radius of the Z n lattice. The n -volumeof the FP and of the WS can be expressed in terms of R ,as V FP = V Z n WS = 2 n n − n/ R n . (2.9)Later, we will compare the properties of different latticesat fixed V W S . B. The A ∗ n lattice The A ∗ n lattice is a classical root lattice, whose attrac-tions have been discussed in detail by [23]. For n ≤
17 itis either the thinnest classical root lattice, or close to thethinnest one. (Note however that thinner non-classicallattices have been constructed numerically, by semidef-inite optimization in the space of lattices. The currentrecord-holders are listed in Table 2 of [24].)The A ∗ n lattice is generated by basis vectors chosen tosatisfy (see, e.g., [22, 25]) e i · e j = (cid:40) i = j − /n for i (cid:54) = j. (2.10)The vectors e i are easily visualized: they point from theorigin to n of the n +1 vertices of an equilateral n -simplex.(The unit vector from the origin to the final vertex of thesimplex is − e − · · · − e n , which implies that the centerof the simplex lies at the origin of coordinates.)For this lattice, the distance function Eq. (2.1) is r = n (cid:88) i =1 ∆ y i − n n (cid:88) i,j =1( i (cid:54) = j ) ∆ y i ∆ y j , = (cid:18) n (cid:19) n (cid:88) i =1 ∆ y i − n (cid:18) n (cid:88) i =1 ∆ y i (cid:19) , (2.11)and the metric is g ij = − /n · · · − /n − /n · · · − /n ... ... . . . ... − /n − /n · · · . (2.12)Using recursion and row reduction, or applyingSylvester’s theorem, it is easy to see that the determi-nant is g = n − n ( n + 1) n − . (2.13)From Eq. (2.2), one obtains V FP = n − n/ ( n + 1) ( n − / (cid:96) n (2.14)for the n -volume of the FP.We now compute the covering radius R , which is thedistance from the origin to the most distant point of theWS centered at the origin. To find the boundary of theWS centered at the origin, we first find the equation ofthe plane that lies halfway between the origin and a lat-tice point with coordinates (0 , · · · , , (cid:96), · · · , (cid:96) ), where thenumber of zeros is k and the number of (cid:96) ’s is n − k .We take this form for an FP vertex because it is suffi-ciently general, i.e. according to the distance functionform Eq. (2.11), the coordinates can be permuted with-out changing the distance value. In contrast to the Z n lattice, every FP vertex defines a WS boundary planes.After multiplying the squared distance by an overall fac-tor of n/ ( n + 1), the coordinates in the planes satisfy the equation n (cid:88) i =1 y i − n + 1 (cid:18)(cid:88) i y i (cid:19) = k (cid:88) i =1 y i + n (cid:88) i = k +1 ( y i − (cid:96) ) − n + 1 (cid:18) k (cid:88) i =1 y i + n (cid:88) i = k +1 ( y i − (cid:96) ) (cid:19) . (2.15)This expression can be simplified, rearranged, and di-vided by 2 (cid:96) ( n − k ) / ( n + 1) to obtain n (cid:88) i =1 y i = n + 1 n − k n (cid:88) i = k +1 y i − k + 12 (cid:96) . Writing the (unity) coefficient of the l.h.s. as ( n +1) / ( n − k ) − ( k + 1) / ( n − k ), and canceling the common terms inthe sums, gives n + 1 n − k k (cid:88) i =1 y i − k + 1 n − k n (cid:88) i =1 y i = − k + 12 (cid:96) . Multiplying this expression by ( n − k ) / ( k + 1) yields thefollowing formula, which defines the planes bounding theWS cell: n (cid:88) i =1 y i = n + 1 k + 1 k (cid:88) i =1 y i + n − k (cid:96) . (2.16)Although we obtained this equation for a specific subsetof vertices, it is trivial to obtain the corresponding equa-tion for any vertex, by replacing the sum from 1 to k with a sum over any k of the coordinates. Changing thesign of (cid:96) gives the corresponding parallel plane boundingthe WS on the other side of the origin. For this reason,the WS is sometimes called a “permutohedron” [22] anddenoted P n .To obtain the covering radius R , we intersect a set of n bounding planes defined by Eq. (2.16), to identify a pointat this radius in the WS. The k = 0 equation implies n (cid:88) i =1 y i = n(cid:96) . (2.17)The k = 1 equation then implies y = (cid:96)/ ( n +1). Combin-ing these with the k = 2 equation implies y = 2 (cid:96)/ ( n +1).Continuing in this fashion, intersecting all of the planesimplies y i = i(cid:96)/ ( n + 1). The squared covering radius ofthe WS is thus given by R = (cid:18) n (cid:19) n (cid:88) i =1 y i − n (cid:18) n (cid:88) i =1 y i (cid:19) = (cid:18) n (cid:19) (cid:96) ( n + 1) (1 + · · · + n ) − n (cid:18) n(cid:96) (cid:19) = (cid:96) n ( n + 1) n ( n + 1)(2 n + 1)6 − n(cid:96)
4= 112 ( n + 2) (cid:96) . (2.18)As before, we can express the WS n -volume in terms of R : V A ∗ n WS = (cid:20) n + 1) n ( n + 2) (cid:21) n/ ( n + 1) − / R n . (2.19)This will be useful later, when we compare lattices atfixed WS volume. III. THE FRACTION OF LOST DETECTIONS
A template bank is discrete, so most points in param-eter space do not have an exactly matching template. Asa result, there is the detection mismatch, which, on aver-age, results in lost detections. Here, N D denotes the totalnumber of sources detectable above a certain SNR thresh-old, and N lost is the number of lost detections, in com-parison with a closely spaced (ideal) bank that catchesall signals.The fraction of lost detections depends upon the effec-tive dimensionality d of the source distribution. If sourcesare uniformly placed in a 3-dimensional Euclidean space,then the number of sources N grows as the distance L as dN ∝ L dL . Similarly, if they are arranged in a 2-dimensional plane (for example, a thin Galactic disk)then dN ∝ LdL . So here we define d by dN ∝ L d − dL and assume that the squared SNR is proportional to1 /L .If the volume of parameter space is much larger thana WS cell and the template bank is a lattice, then thefraction of lost detections in the spherical approximation[21] is given by [19] N lost N D ≈ V WS (cid:90) WS f ( r ) dV, (3.1)where the integral is over a single WS cell, and the inte-grand is f ( r ) = (cid:40) − cos d r for r ≤ π/
21 for r > π/ . (3.2)The ratio N lost /N D defines the “loss fraction” of the lat-tice, i.e. the fraction of potentially-detectable signalswhich the lattice fails to catch. Equivalently, 1 − N lost /N D is the efficiency of the lattice: the expected fraction ofpotentially detectable signals which are indeed found.Provided that the WS cell is not too large, so that R < π/
2, the integrand can be expanded in a series,giving a loss fraction N lost N D ≈ V WS (cid:90) WS (cid:0) − cos d ( r ) (cid:1) dV = 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 (105 d − d + 588 d − (cid:104) r (cid:105) + · · · . (3.3) Here, (cid:104) r p (cid:105) = 1 V WS (cid:90) WS r p dV (3.4)denotes the normalized p ’th moment of the lattice.Provided that the effective dimensionality of the sourcedistribution d > /π ≈ .
81, the quadratic approxima-tion always implies a larger fraction of signals lost thanthe spherical approximation, because 1 − cos d ( r ) < r d/ r ∈ [0 , π/ Z n lattice. The first six of these, whichsuffice for this paper, are (cid:104) r (cid:105) = n(cid:96) , (cid:104) r (cid:105) = n(cid:96)
720 (5 n + 4) , (cid:104) r (cid:105) = n(cid:96) n + 84 n + 16) , (3.5) (cid:104) r (cid:105) = n(cid:96) n + 840 n + 656 n − , (cid:104) r (cid:105) = n(cid:96) n + 3080 n + 5456 n + 352 n − , (cid:104) r (cid:105) = n(cid:96) n + 2102100 n + 6646640 n + 3747744 n − n + 35328) . We note that all of these quantities can be re-expressedin terms of the covering radius R = n(cid:96) /
4. The corre-sponding even moments for the A ∗ n lattice are computedin Appendix B, but not repeated here.In the following we shall consider d = 2 and d = 3dimensional source distributions. A. The Z n lattice: d = 2 case For a source distribution with effective dimensionality d = 2, we now evaluate the fraction of lost sources, as-suming that the covering radius R ≤ π/
2. The integrandof Eq. (3.3) ( f ( r ) = sin r , the mismatch in the spheri-cal approximation [21]) is approximated (within 1% ) bytaking terms up to the eighth moment. Then, Eq. (3.3)takes the form N lost N D ≈ (cid:104) r (cid:105) − (cid:104) r (cid:105) + 245 (cid:104) r (cid:105) − (cid:104) r (cid:105) = 13 R − n + 4135 n R + 70 n + 168 n + 3242525 n R − n + 840 n + 656 n − n R . (3.6)We plot this quantity in Fig. 2(a), where m worst = sin R denotes the worst-case mismatch in the spherical approx-imation. ( a ) ( b )FIG. 2. The fraction of lost detections for the Z n lattice in dimensions n = 2 (lower solid curve) and n = 4 (upper solid curve).For larger n the corresponding curves group very close together; the red dashed curve shows the n → ∞ limit of Eq. (4.1). Theleft-hand plot shows a d = 2 dimensional source distribution and the right-hand plot shows a d = 3 dimensional distribution.The fraction of lost detections depends upon the spacing of the template bank, which is set by the covering radius R ; in thespherical approximation [21], the worst mismatch m worst = sin R . For closely spaced templates (small R ) no detections arelost. For comparison the quadratic approximation Eq. (3.7) is shown as a dotted curve. It predicts more lost signals thanthe spherical approximation suggests. The fraction of lost detections at maximum mismatch R = π/ n → ∞ and at covering radius R = π/
2, about 62% of detections are lost for a d = 2 dimensional source distribution, andabout 77% are lost for a d = 3 dimensional source distribution.TABLE I. The maximal fraction of lost detections N lost /N D for the Z n and A ∗ n lattices in small dimensions n , for d = 2 and d = 3 dimensional source distributions. Note that all latticeshave “maximal” WS radius R = π/
2, which means that at afixed dimension n , the WS cells have smaller volume for Z n than for A ∗ n . Z n Z n A ∗ n A ∗ n n d = 2 d = 3 d = 2 d = 32 0.558 0.665 0.642 0.7363 0.579 0.697 0.720 0.8164 0.589 0.714 0.771 0.8635 0.595 0.724 0.806 0.8936 0.599 0.731 0.832 0.9147 0.602 0.736 0.851 0.9288 0.605 0.740 0.867 0.9399 0.606 0.743 0.880 0.94810 0.608 0.745 0.890 0.95411 0.609 0.747 0.899 0.96012 0.610 0.749 0.906 0.964 n → ∞ Fig. 2(a) also compares the spherical approximation[21] to the mismatch with the prediction one would findusing the normal quadratic approximation. If the lat-tice is widely spaced (sparse), then the spherical approx-imation predicts significantly fewer lost signals than thestandard quadratic approximation. The quadratic ap- proximation keeps only the first term in Eq. (3.6), so (cid:20) N lost N D (cid:21) Quadratic-Approximation = 13 arcsin ( √ m worst ) , (3.7)which is valid in any dimension n . To enable a fair com-parison with the spherical approximation, we need to ex-amine the two expressions for the same lattice, meaningat the same WS radius R . So in Eq. (3.7), this is stillrelated to the worst-case mismatch via the spherical ap-proximation [21] m worst = sin R (rather than with thequadratic approximation m worst = R ).Results of numerical computations of the maximalfraction of lost detections are presented in Table I. B. The Z n lattice: d = 3 case For d = 3 the integrand in Eq. (3.3) is f ( r ) = 1 − cos r ,and we again assume R ≤ π/
2. The expression Eq. (3.3)takes the following form: N lost N D ≈ (cid:104) r (cid:105) − (cid:104) r (cid:105) + 61240 (cid:104) r (cid:105) − (cid:104) r (cid:105) + 703172800 (cid:104) r (cid:105) − (cid:104) r (cid:105) . (3.8)Here, to maintain 1% accuracy in the integrand we havehad to include more terms than for d = 2. Fig. 2(b) ( a ) ( b )FIG. 3. The fraction of lost detections for the A ∗ n lattice in different numbers of dimensions (moving upwards) n =2 , , , , , d = 2 dimensional source distribution and the right-hand plots shows a d = 3 dimensional distribution. As in the previous figure, the n → ∞ limit is shown in red, and the plot is restricted to R ≤ π/
2. The quadratic approximation is shown as a dotted curve in the limit n → ∞ [see Eq. (4.5)]. The fraction of lostdetections at maximum mismatch R = π/ illustrates how the fraction of lost detections depends onthe covering radius (via the worst-case mismatch m worst ).In the case of quadratic approximation we keep only thefirst term in Eq. (3.8), (cid:20) N lost N D (cid:21) Quadratic-Approximation = 12 arcsin ( √ m worst ) , (3.9)valid in any dimension n . For a widely spaced latticethe spherical approximation predicts significantly fewerlost signals than the standard quadratic approximation.The worst-case values (fraction of lost detections at WSradius R = π/
2) are shown in Table I.
C. The A ∗ n lattice: d = 2 and d = 3 cases As for the Z n lattice, we can again estimate how thefraction of lost detections depends upon the covering ra-dius. For the A ∗ n lattice, we can compute the moments (cid:104) r p (cid:105) exactly, but cannot give a closed analytic form aswe did for the Z n lattice. We use the exact expressionsobtained in Appendix B, and substitute these into theexpressions Eq. (3.6) and Eq. (3.8). The plots of the frac-tion of lost detections versus m worst are given in Fig. 3,and some worst-case values are shown in Table I. IV. LARGE n LIMITS
The reader will notice that as the dimension n of theparameter space gets large, the curves appear to ap- proach a limit. This is explained in Sec. VI, where weshow that as n gets large, the mismatch distribution func-tion becomes sharply peaked at r = R / Z n lattice and at r = R for the A ∗ n lattice. Thus, for the Z n lattice, Eq. (3.3) immediately giveslim n →∞ N lost N D = 1 − cos d (cid:16) R/ √ (cid:17) = 1 − cos d (cid:18) arcsin √ m worst √ (cid:19) , (4.1)where we have used the relationship m worst = sin R be-tween the WS radius and the worst-case mismatch.For a source distribution with effective dimensionality d = 2 this has a limiting value of N lost /N D ≈ .
620 for m worst = 1. So, if there are at least a few dimensionsto parameter space, then placing templates in a rectan-gular grid at unit mismatch will recover about 38% ofsignals. For a source distribution with effective dimen-sionality d = 3, the limiting value is N lost /N D ≈ . A ∗ n lattice, Sec. VI shows that in thelimit of large n we havelim n →∞ (cid:104) r (cid:105) WS = R , (4.2)where the covering radius R ∈ (0 , π/ n →∞ (cid:104) r m (cid:105) WS = R m , m = 1 , , , ... . (4.3)Thus, lim n →∞ N lost N D = 1 − cos d R , (4.4)which leads to a worst-case limit of unity, as shown in Ta-ble I. For large dimensions the quadratic approximationof the fraction of lost detections can also be constructedin the closed form [cf. Eq. (3.7)],lim n →∞ (cid:20) N lost N D (cid:21) Quadratic-Approximation = d ( √ m worst ) . (4.5)This expression is shown by the dotted curves in Fig. 3. V. COMPARISON OF Z n AND A ∗ n AT FIXEDCOMPUTING COST
To evaluate the relative loss fractions of the Z n and A ∗ n lattices at fixed computing cost, we must compare themfor identical values of the WS cell volume V WS . This en-sures that the same number of templates would be em-ployed to cover a given volume of parameter space.Such a comparison is shown in Fig. 4. The horizontalaxis “ x ” in these plots is proportional to the “squaredlength” V /n WS . In the figure, this is normalized to reachunity when the covering radius of the A ∗ n lattice reaches R = π/
2. From Eq. (2.19), the resulting normalizationfactor is the inverse of V /n WS-Max = (cid:20) V A ∗ n WS ( R = π/ (cid:21) /n = 3 π ( n + 1) n ( n + 2) ( n + 1) − /n . (5.1)Thus, if we denote the horizontal axes of Fig. 4 by x =( V WS /V WS-Max ) /n , by using Eq. (2.9) and Eq. (2.19) wehave x = π (cid:18) n + 2 n + 1 (cid:19) (1 + n ) /n R for Z n R π for A ∗ n , (5.2)where R is WS cell covering radius of the correspondinglattice. Note that when the two lattices are comparedat a given point on the x -axis, they have equal WS cellvolume, hence they have different WS radii, and corre-spondingly different values of (cid:96) .At fixed V WS , the WS radius R of the Z n lattice isalways larger than the WS radius of the A ∗ n lattice. Sincewe allow the WS radius for A ∗ n to reach maximal value π/
2, it follows that in the plots in Fig. 4, the WS radius of Z n exceeds π/ Z n reaches R = π/ Z n lattice is set to unity for r > π/ Z n results have been obtained with Monte Carlointegration, since the analytic formulae obtained earlieronly hold for R ≤ π/
2. As can be seen from Eq. (5.2),the location of this dot approaches x = 1 / n limit.One can see that these plots have taken us away fromthe quadratic approximation to the mismatch. To getsome sense of how far away, consider the maximum mis-match at the locations of the dots. In the quadratic ap-proximation, this would be m = r = π / ≈ .
47, morethan double the maximum allowed value of m = 1. Inthe quadratic approximation to the mismatch, the lowercurves of Fig. 4 would be straight lines tangent to thegiven curves at V WS = 0. The upper curves would behorizontal lines passing through the V WS = 0 values.The results of [19] show that for small mismatch, wherethe quadratic approximation applies, the A ∗ n lattice isonly slightly less lossy than the Z n lattice. We can nowsee that this marginal advantage decreases for larger mis-match: the upper part of Fig. 4 shows the ratio of theloss fractions for the two lattices. The efficiency of the A ∗ n lattice is at most ≈
10 % higher than that of the Z n lattice.The large- n limits of Sec. IV are informative and canbe easily evaluated. Taking n → ∞ in Eq. (5.2) the lossfractions Eq. (4.1) and Eq. (4.4) for both the lattices takethe identical form N lost N D = 1 − cos d (cid:16) π √ x (cid:17) . (5.3)This is shown by the dotted red curves in Fig. 4. Thetransition point x = 1 / Z n lattice equal to π/ A ∗ n lattice has the same curve, the transition isonly relevant for the Z n lattice. For large n , the ratio ofthe loss fractions approaches unity, as can be seen fromEq. (5.3). VI. DISTRIBUTION FUNCTION OF THESQUARED DISTANCE
To understand and interpret the results presentedabove, it is helpful to define the mismatch distributionfunction P m ( m ). This is defined as a probability dis-tribution: if points in parameter space are chosen “atrandom” then the probability that the mismatch lies inthe range ( m, m + dm ) is P m ( m ) dm . Here, we compute P m ( m ) under the assumption that the probability of se-lecting a particular point in parameter space is a uniformdistribution in the lattice coordinates y i ∈ [0 , (cid:96) ]. This isequivalent to a uniform distribution in x i .In the quadratic and spherical approximations [21], themismatch is a one-to-one function of the squared distance r , assuming of course in the spherical case that we re-strict attention to r ∈ [0 , π/ N l o s t , Z / N l o s t , A Source Dimension 20.00 0.25 0.50 0.75 1.00( V WS / V WS Max ) n N l o s t / N d e t e c t e d A * nn n n=2n=4n=6n=10n=20n=100 1.001.051.10 N l o s t , Z / N l o s t , A Source Dimension 30.00 0.25 0.50 0.75 1.00( V WS / V WS Max ) n N l o s t / N d e t e c t e d A * nn n n=2n=4n=6n=10n=20n=100 ( a ) ( b )FIG. 4. A comparison of lattice loss fractions at fixed computing cost. Lower curves: the loss fractions N lost /N D for the Z n and A ∗ n lattices, at fixed WS cell volume V WS . The red dotted curves represent the lattice loss fraction in the limit n → ∞ [seeEq. (5.3)]. Upper curves: ratios of these loss fractions. The horizontal axis normalization is given by Eq. (5.1). The dots onthe curves indicate where the covering radius for Z n reaches R = π/
2. The left (right) plots are for a d = 2 ( d = 3) dimensionalsource distributions. function P r ( r ), assuming the same uniform distribu-tion of the y i . This distribution function can be used tocompute an average value of an integrable function f of r , (cid:104) f ( r ) (cid:105) = 1 V WS (cid:90) WS f ( r ) dV = (cid:90) R f ( r ) P r ( r ) dr . (6.1)Thus, the quantity we wish to compute is the distributionof the values of the quadratic forms given in Eq. (2.5) forthe Z n lattice and in Eq. (2.11) for the A ∗ n lattice. A. r -distribution for the Z n lattice For finite values of the dimension n we have not founda simple closed form for P r ( r ), although we can giveexpressions for n = 1 ,
2, and 3. However, the large- n limit is easily computed. To compute the radius distribution function P r ( r ) forlarge n , we make use of the central limit theorem [26].Consider the distance Eq. (2.5). In the large- n limit it isthe sum of many independent random variables, each ofwhich has the same distribution. Thus, we expect thatit should approach a normal or Gaussian distribution,characterized entirely by the mean and variance of thedistribution.We have already calculated the moments of r for the Z n lattice. The mean and variance are given by (cid:104) r (cid:105) = 112 n(cid:96) = 13 R , (6.2)and σ = (cid:104) r (cid:105) − (cid:104) r (cid:105) = 1180 n(cid:96) = 445 n R . (6.3)From these, the large- n limit follows immediately. Notethat as n gets large, the variance vanishes, which meansthat the distribution becomes sharply peaked.0 r / R D i s t a n c e d i s t r i b u t i o n P r ( r ) n lattice n=2n=4n=6n=8n=10 0.0 0.2 0.4 0.6 0.8 1.0 r / R D i s t a n c e d i s t r i b u t i o n P r ( r ) A * n latticen=2n=4n=6n=8n=100.1 0.2 0.3 0.4 0.5 0.6 r / R D i s t a n c e d i s t r i b u t i o n P r ( r ) n lattice n=20n=50n=100n=200n=500 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r / R D i s t a n c e d i s t r i b u t i o n P r ( r ) A * n latticen=20n=50n=100n=200n=500n=1000 FIG. 5. The probability distribution of the squared distance P r ( r ) is shown as the solid curves for the Z n (left plots) and A ∗ n (right plots) lattices, for a varying number of dimensions n . The top curves show small numbers of dimensions n = 2 , , , , n = 20 , , , , Z n lattice, as expected from the central limit theorem, the Gaussian approximation approaches the true distribution as n → ∞ , which is a Dirac delta function peaked at r /R = 1 /
3. For A ∗ n , the central limit theorem does not apply, and theGaussian approximation does not approach the true distribution for large dimension. Nevertheless, as n → ∞ , the distributionsapproaches a Dirac delta function peaked at r /R = 1. If n is large enough that the central limit theorem ap-plies, then the distribution of squared distance is a Gaus-sian normal distribution P r ( r ) dr = (2 πσ ) − / e − ( r − R / / σ dr . (6.4)Note that if the dimension n is large, then this has vanish-ing support for negative r , otherwise the normalization may be suitably adjusted.In the n → ∞ limit with fixed mismatch, the variancevanishes, and the distribution approaches a Dirac deltafunction lim n →∞ P r ( r ) = δ (cid:18) r − R (cid:19) . (6.5)1 Dimension n10 -3 -2 -1 › r fi σ = › r fi − › r fi A ∗ n lattice R FIG. 6. The mean and variance of the squared radius for the A ∗ n lattice for dimensions from 1 to 1000, obtained exactlyusing the recursion in Appendix B. At large dimension thedistribution is a narrow peak at the squared WS radius R . In Fig. 5 we show how this limit is approached. When n is larger than 2, one has 2 σ < ( R / and as soon as n is a few times larger than this, the Gaussian distributionbecomes a good approximation to the actual mismatch. B. r -distribution for the A ∗ n lattice The case of the A ∗ n lattice is not as simple. The squareddistance is still a quadratic form which can be diagonal-ized, but the variables which make it up are no longerindependent, because they are constrained by the bound-aries of the WS. It is unlike the Z n lattice, where theseconstraints are independent for each variable. Hence, thecentral limit theorem cannot be applied.It is informative to examine the moments of r definedby Eq. (3.4), which are computed exactly via recursionin Appendix B. Fig. 6 shows the mean and variance of r for the A ∗ n lattice. One immediately sees a signifi-cant difference when compared with the Z n lattice: atlarge dimension, the mean value of squared radius (cid:104) r (cid:105) approaches the squared WS radius R , whereas for Z n it is 1 / (cid:104) r m (cid:105) are shown in Fig. 7: for large n they asymptote to R m .It is straightforward to study the distribution functionnumerically. First, select points at random from withinthe FP, by drawing the lattice coordinates y , · · · , y n from independent uniform distribution in the range [0 , (cid:96) ].Then identify the closest lattice point to x = y i e i and cal-culate the distance between the two. We now describe FIG. 7. The even moments (cid:104) r m (cid:105) for m = 2 , , , , ,
12 inthe units of R m for the A ∗ n lattice for dimensions from 1 to1000 are shown in the ascending order. how to identify this closest lattice point. (An algorithmis given in [22] for A n as well as the correspondence withthe dual lattice A ∗ n , but we were unable to implement it.)It is straightforward to show that the closest latticepoint to x must be one of the vertices of the FP. Sincethere are 2 n such vertices, when n is large, it’s not compu-tationally feasible to check the distances to all of them.However, it is trivial to show that the distance to theclosest lattice point is unchanged if we permute the or-dering of the lattice coordinates y i . So the first step ofsimplification is to reorder the lattice coordinate valuesof y i in increasing order.We now prove the following. If 0 ≤ y ≤ · · · ≤ y n ≤ (cid:96) are the lattice coordinates of a point in the FP,then the closest FP vertex has coordinates of the form(0 , · · · , , (cid:96), · · · , (cid:96) ), where there are k zeros followed by( n − k ) (cid:96) ’s. The proof is by contradiction.Suppose that the closest vertex to the point with lat-tice coordinates ( y , · · · , y n ) is a point with lattice co-ordinates A = (0 , · · · , , (cid:96), , (cid:96), · · · , (cid:96) ) and is at squareddistance r A . We use y R to denote the lattice coordinatevalue at the position of the rightmost zero, and y L todenote the value at the leftmost (cid:96) . Now, construct a dif-ferent lattice vertex B, by swapping the leftmost (cid:96) withthe 0 just to its right, so that B = (0 , · · · , , (cid:96), (cid:96), · · · , (cid:96) ),and denote its squared distance from y by r B . The dif-ference between the squared distances is r A − r B = (cid:18) n (cid:19) (cid:18) ( y L − (cid:96) ) + y R − ( y R − (cid:96) ) − y L ) (cid:19) = 2 (cid:96) (cid:18) n (cid:19) ( y R − y L ) . (6.6)Since the coordinates are ordered so that y L < y R , it2follows that r A − r B > A is not theclosest lattice vertex to y . The same argument showsthat swapping a leftmost (cid:96) with a 0 anywhere to its rightwill always decrease the distance. The result follows byinduction.This makes it computationally straightforward to iden-tify the closest vertex to any point inside the FP. First,sort the lattice coordinates in increasing order. Then, cal-culate the distances to the n +1 vertices with coordinatesof the form (0 , · · · , , (cid:96), · · · , (cid:96) ) and select the minimum.We have used this method to find P r ( r ) numericallyfor the A ∗ n lattice, for dimensions from n = 1 to n = 1000.This is plotted in Fig. 5. In comparison with the cu-bic lattice Z n , two differences are immediately apparent.The first is that as the dimension n increases, the dis-tribution increasingly becomes peaked around the WSradius R , and the second is that the Gaussian approxi-mation (with the correct mean and variance) is not good,because it does not fall off fast enough as r → R . VII. CONCLUSION
In this paper, we have computed and compared theloss fractions of two template grids. The first is basedon the simple cubic lattice Z n , and the second is basedon the root lattice A ∗ n , which is a generalization of thetwo-dimensional hexagonal lattice. In particular, we ex-tend the results of [19] to the case of large mismatch, byexploiting the spherical approximation [21].The main result is rather clear, and visible in the upperpart of Fig. 4. The slight advantages offered by the A ∗ n lattice at small mismatch decrease at larger mismatch.This can be easily understood from the distribution of thesquared radius for points randomly selected within theWigner-Seitz (WS) cell. As the dimension n of parameterspace increases, this distribution becomes an increasinglynarrow peak centered closer and closer to the squared WSradius.We believe that this behavior may be general, and truefor any lattice in the limit as the dimension n → ∞ . Tostate it precisely, the distribution function for the squaredradius becomes an increasingly narrow peak, which istrue if and only if lim n →∞ (cid:104) r m (cid:105) = (cid:104) r (cid:105) m , (7.1)with the understanding that the WS radius R is heldfixed during the limiting process. We have tried to provethis using Jensen’s inequality [27], but are not convincedthat our argument is correct.The final messages for the data analyst are simple ones.First, a fairly effective template-based search can be con-structed at mismatch values that are shockingly highin the quadratic approximation (quadratic mismatch ex-ceeding unity!). Second, if the goal is to detect as manysignals as possible at fixed computing cost, there is littlemotivation for using template banks based on sophisti- cated lattices such as A ∗ n . These offer only minimal bene-fit when compared with the humble cubic lattice Z n , andthat minor advantage diminishes as the template separa-tion increases. VIII. ACKNOWLEDGMENTS
We thank Mathieu Dutour Sikiri´c for bringing thethinnest known lattices of [24] to our attention.
Appendix A: Even moments of the Z n lattice For the Z n lattice, the general even-order moment canbe computed as follows. One uses the multinomial ex-pansion to write (cid:104) r m (cid:105) = (cid:88) k + ··· + k n = m (cid:18) mk , · · · , k n (cid:19) n (cid:89) i =1 (cid:104) x k i i (cid:105) , (A1)where the sum is over all non-negative integer k i whosesum equals m . The multinomial coefficient is (cid:18) mk , k , · · · , k n (cid:19) = m ! k ! k ! · · · k n ! , (A2)and the coordinate moments are (cid:104) x k (cid:105) = 1 (cid:96) (cid:90) (cid:96)/ − (cid:96)/ x k dx = 12 k + 1 (cid:18) (cid:96) (cid:19) k . (A3)In the sum Eq. (A1), there are many identical terms onthe r.h.s. which are obtained by permutation of the in-dices of the k i . The number of these identical terms de-pends upon the number of distinct non-zero values takenby the k i , which in turn depends upon the dimension n .Suppose that for each term, the non-zero k i are sortedin increasing order; there at most m of them. Let q ≤ m denote the number of these non-zero k i , and let n denotethe number of k i which have the smallest value, n thenext smallest, and so on; the sum is bounded by (cid:80) i n i ≤ m . Then the number of equivalent (under permutation)terms which appear on the r.h.s. of Eq. (A1) is equal tothe number of ways in which n coordinates can be chosenfrom the n , and n can be chosen from the remaining n − n , and so on. This is N ( k , · · · , k q )= (cid:18) nn (cid:19)(cid:18) n − n n (cid:19) × · · · × (cid:18) n − n − · · · − n p − n p (cid:19) = n ! n ! n ! · · · n p !( n − n − · · · − n p )! , (A4)where the quantities in the second line are the standardbinomial (choice) coefficients; the r.h.s. is a polynomialin n of order ≤ m . Thus one obtains (cid:104) r m (cid:105) = (cid:88) k + ··· + k q = m (cid:0) mk , ··· ,k q (cid:1) N ( k , · · · , k q )(2 k + 1) · · · (2 k q + 1) (cid:18) (cid:96) (cid:19) m , (A5)3where the sum is over all distinct (under permutation)partitions k i .For example, for m = 5, the r.h.s. of Eq. (A5)has seven terms, with the following sets of k i : { , , , , } , { , , , } , { , , } , { , , } , { , } , { , } ,and { } . Respectively, these have n i given by { } , { , } , { , } , { , } , { , } , { , } , and { } , with cor-responding N given by n ( n − n − n − n − / n ( n − n − n − / n ( n − n − / n ( n − n − / n ( n − n ( n −
1) and n . Thus oneobtains (cid:18) (cid:96) (cid:19) − (cid:104) r (cid:105) = (cid:18) , , , , (cid:19) n ( n − n − n − n − + (cid:18) , , , (cid:19) n ( n − n − n − (cid:18) , , (cid:19) n ( n − n − (cid:18) , , (cid:19) n ( n − n − (cid:18) , (cid:19) n ( n − · (cid:18) , (cid:19) n ( n − · (cid:18) (cid:19) n . This simplifies, to give the tenth moment of Eq. (3.5).The supplementary materials for this manuscript in-clude a short Mathematica script to calculate arbitraryeven moments of the Z n lattice. Appendix B: Even moments of the A ∗ n lattice Here we give a general expression for computation ofany even moment of the A ∗ n lattice. The computationtechnique is a generalization of Chapter 21 Section 3.Fof [22], where it is used to find the second moment.The un-normalized and normalized p ’th moments of aregion or object D are defined as U p ( D ) = (cid:90) D r p dV, and I p ( D ) = U p ( D ) /U ( D ) , (B1)where D is the domain of integration and the radius r ismeasured from the origin O (see Fig. 8).The WS cell in dimension n is called a permutohedronand is denoted P n . It has a complex shape with ( n + 1)!vertices and 2 n +1 − U ( P n ) is the volume of the WS cell P n . Thenormalized m ’th moment I p ( P n ) is obtained by dividingout the volume.Note that the length conventions used in this Appendixfollow [22], and differ from the conventions used in theremainder of this paper. To transform a quantity associ-ated with P n with dimensions of length d in this Sectioninto the units used in the remainder of the paper, multi-ply by (cid:20) (cid:96) n ( n + 1) (cid:21) d/ . (B2) For example, in the conventions of this Section, the pointin P n most distant from the center has squared radius n ( n + 1)( n + 2) /
12, which should be compared withEq. (2.18), and the volume is U ( P n ) = ( n + 1) n − / ,which should be compared with Eq. (2.14). FIG. 8. The n -dimensional pyramid associated with ( n − F n − ,s . The point O is such that for allcongruent faces F n − ,s the associated pyramids are congruent.The axis Ox is perpendicular to the face F n − ,s , and h n − ,s isthe distance form O to the face. The increment dx is thicknessof the slab at x . Each face of P n is the direct product of a pair of lower-dimensional permutohedrons . For n even there are n/ n odd there are ( n + 1) / s = 0 , · · · , n −
1. A face of type s , F n − ,s , isthe Cartesian product, F n − ,s = P s × P n − s − ; faces oftype s and faces of type n − s − s is the binomial coefficient (cid:18) n + 1 s + 1 (cid:19) . (B3)The squared distance from the center of P n to the centerof a face of type s is h n,s = 14 ( s + 1)( n − s )( n + 1) . (B4)By symmetry, the line from the center of P n to the centerof any face is orthogonal to the face. We call this thecenter line to the face.Because the faces are formed from lower-dimensionalpermutohedrons, the moments may be calculated by re-cursion. We divide P n into generalized pyramids, one A face of P n is only the direct product (metrically as well asgeometrically!) of lower-dimensional faces of P n if we follow thethe “dimension-dependent” length conventions of [22]. P n and extend to any-where in that face. These pyramids are disjoint (apartfrom a set of measure zero on their boundaries) and theirunion is P n . To compute the moments of P n , we computethe moments of the pyramids and sum them.The m ’th moment of each pyramid can be found withelementary calculus. We slice each pyramid into slabs ofthickness dx , where x ∈ [0 , h n,s ] is a a coordinate thatruns along the center line to a face of type s , and theslicing is orthogonal to the center line shown in Fig. 8.Each slab has n -volume dV = x n − h n − n,s U ( P s ) U ( P n − s − ) dx, (B5)so by integration over x the volume of the pyramid is (cid:90) h n,s dV = 1 n h n,s U ( P s ) U ( P n − s − ) . (B6)Summing over all faces gives U ( P n ) = 1 n n − (cid:88) s =0 (cid:18) n + 1 s + 1 (cid:19) h n,s U ( P s ) U ( P n − s − ) . (B7)This recursion relation, together with the initial value U ( P ) = 1, determines the volume U ( P n ) for dimen-sions n > U m ( P n ), m = 0 , , , , ... , we beginwith the expression U m ( P n ) = n − (cid:88) s =0 (cid:18) n + 1 s + 1 (cid:19) U m ( P n,s ) , (B8)where U m ( P n,s )’s are the moments of n -dimensionalpyramids P n,s into which a permutohedron P n is decom-posed. Every such moment can be calculated by usingthe definition Eq. (B1), substituting for r the expression r = x h n,s ρ n − ,s + x , (B9)where ρ n − ,s is the distance from the point of intersectionof the axis Ox with the face F n − ,s to arbitrary point ofthe face (see Fig. 8), as follows: U m ( P n,s ) = (cid:90) F n − ,s dV (cid:90) h n,s dx (cid:18) x h n,s ρ n − ,s + x (cid:19) m = m (cid:88) k =0 (cid:18) mk (cid:19) (cid:90) F n − ,s ρ kn − ,s dV (cid:90) h n,s x m + n − h n − kn,s dx , where the volume element dV is in the face F n − ,s . (Forodd moments m = k + 1 /
2, where k = 0 , , , ... the finitesum in the expression above is replaced by an infiniteseries.) Using the definition Eq. (B1) for the moments U k ( F n − ,s ) of faces F n − ,s (here, with origin at the cen-ter of the face) and integrating over x we obtain U m ( P n,s ) = h m − k )+1 n,s n + 2 m m (cid:88) k =0 (cid:18) mk (cid:19) U k ( F n − ,s ) . (B10)Substituting this expression into Eq. (B8) we obtain U m ( P n ) = 1 n + 2 m n − (cid:88) s =0 m (cid:88) k =0 (cid:18) n + 1 s + 1 (cid:19)(cid:18) mk (cid:19) × h m − k )+1 n,s U k ( F n − ,s ) . (B11)The next step is to consider the face F n − ,s as the Carte-sian product P s × P n − s − and apply again the definitionEq. (B1) to the moments U k ( F n − ,s ), with the origin atthe center of the face. Replace r with ρ n − ,s = ρ s + ρ n − − s , (B12)and use the binomial theorem to raise Eq. (B12) to power k . Employing the definition Eq. (B1) for the moments U j ( P s ) and U k − j ) ( P n − − s ), we obtain U k ( F n − ,s ) = k (cid:88) j =0 (cid:18) kj (cid:19) U j ( P s ) U k − j ) ( P n − s − ) . (B13)Finally, substituting Eq. (B13) into Eq. (B11), we obtainthe following relation for the even moments of P n : U m ( P n ) = 1 n + 2 m n − (cid:88) s =0 m (cid:88) k =0 k (cid:88) j =0 (cid:18) n + 1 s + 1 (cid:19)(cid:18) mk (cid:19)(cid:18) kj (cid:19) × h m − k )+1 n,s U j ( P s ) U k − j ) ( P n − s − ) . (B14)This recursion relation, together with the initial values U ( P ) = 1 and U m ( P ) = 0, for m = 1 , , , ... , definesan arbitrary even-order moment.In Tables II and III we give numerical and exact val-ues for the even moments (cid:104) r m (cid:105) obtained from U m ( P n ),for dimensions n <
16. The un-normalized momentsare computed using the recursion relation Eq. (B14).The normalized moments I m ( P n ) are then defined byB1. Both of these follow the conventions of Conway andSloane, Chapter 21 Section 3F [22]. They are then re-scaled following Eq. (B2) with d = 2 m to obtain (cid:104) r m (cid:105) ,which are in the conventions used everywhere else in thispaper.The following lines of Mathematica are sufficient to cal-culate the arbitrary even moments U m ( P n ) = U[m , n] upto dimensions of several thousand. 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Perlman, “Jensen’s Inequality for a Convex Vector-valued Function on an Infinite-dimensional Space,” Jour-nal of Multiplicative Analysis , 52 (1974). n (cid:104) r (cid:105) /R (cid:104) r (cid:105) /R (cid:104) r (cid:105) /R (cid:104) r (cid:105) /R (cid:104) r (cid:105) /R (cid:104) r (cid:105) /R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P n of the A ∗ n lattice for dimensions n = 1 , . . . , R is given in Eq. (2.18). The exact values of these moments are given in Table III. In the text weargue that lim n →∞ (cid:104) r (cid:105) /R = 1, and that lim n →∞ (cid:104) r m (cid:105) = (cid:104) r (cid:105) m . Hence, as n → ∞ , all of these table entries should approachunity. T A B L E III . E x a c t v a l u e s o f t h e l o w e s t - o r d e r e v e n m o m e n t s f o r t h e A ∗ n l a tt i c e , a s g i v e nb y t h e r e c u r s i o n r e l a t i o n s h i pd e r i v e d i n A pp e nd i x B . N u m e r i c a l v a l u e s m a y b e f o und i n T a b l e II . n (cid:104) r (cid:105) / R (cid:104) r (cid:105) / R (cid:104) r (cid:105) / R (cid:104) r (cid:105) / R (cid:104) r (cid:105) / R (cid:104) r (cid:105) / R15