aa r X i v : . [ m a t h . M G ] A ug Non-uniform packings
Lee-Ad Gottlieb a , Aryeh Kontorovich b, ∗ a Ariel University, Ariel, Israel b Ben-Gurion University of the Negev, Beer Sheva, Israel
Abstract
We generalize the classical notion of packing a set by balls with identical radii tothe case where the radii may be different. The largest number of such balls thatfit inside the set without overlapping is called its non-uniform packing number .We show that the non-uniform packing number can be upper-bounded in termsof the average radius of the balls, resulting in bounds of the familiar classicalform.
Keywords: packing, normed space, metric space
1. Introduction
Packing numbers (along with their dual notion of covering numbers) providea quantitative notion of compactness for a totally bounded metric space andmake a pervasive appearance in empirical processes [1], learning theory [2], andinformation theory [3], among other fundamental results. We note in passingthat violating the triangle inequality destroys the covering-packing duality, andpacking numbers emerge as the more fundamental notion, at least in a learning-theoretic setting [4].We refer the reader to [5] for basic metric-space notions such as total bound-edness and compactness. Briefly, a metric space (Ω , ρ ) is a set endowed with apositive symmetric function, which additionally satisfies the triangle inequality. ∗ Corresponding author
Email addresses: [email protected] (Lee-Ad Gottlieb), [email protected] (AryehKontorovich)
Preprint submitted to Journal of L A TEX Templates August 5, 2020 or r >
0, a set A ⊆ Ω is said to be r -separated if ρ ( a, a ′ ) > r for all distinct a, a ′ ∈ A . The r -packing number of Ω, which we denote by M ( r ), is the max-imum cardinality of any r -separated subset of Ω (and is finite whenever Ω istotally bounded).We will also need the notion of the doubling dimension of a metric space; thelatter is known to be of critical algorithmic [6, 7, 8, 9, 10] and learning-theoreticimportance [11, 12, 13, 14, 15]. Denote by B ( x, r ) = { x ′ ∈ Ω : ρ ( x, x ′ ) ≤ r } the(closed) r -ball about x . If there is a D < ∞ such that every r -ball in Ω iscontained in the union of some D r/ , ρ ) is said to be doubling . Its doubling dimension is defined as ddim(Ω) = ddim(Ω , ρ ) =: log D ∗ ,where D ∗ is the smallest D verifying the doubling property. It is well-known[6, 14] that M ( r ) ≤ (cid:18) r (cid:19) ddim(Ω) , r > , (1)where diam(Ω) = sup x,x ′ ∈ Ω ρ ( x, x ′ ). Further, (1) is tight, as witnessed by theexample of n equidistant points, with r as (1 − ε ) times their common distance,for ε arbitrarily small; in this case, ddim(Ω) = log n .We now refine the notion of r -separated sets to take the individual inter-point distances into account. For A ⊆ Ω and R : A → (0 , ∞ ), we say that A is R -separated if for all a ∈ A ,inf a ′ ∈ A \{ a } ρ ( a, a ′ ) > R ( a ) . (2)In words, for each a ∈ A , its closest neighbor in A is at least R ( a )-away. Theuniform special case R ( a ) ≡ r recovers the classical notion of r -separation.We are now ready to state our main result: Theorem 1.1. If (Ω , ρ ) is a doubling space and A ⊆ Ω is finite and R -separated,then | A | ≤ (cid:18) A )¯ r (cid:19) min { ddim( A ) , ddim(Ω) } , where ¯ r := | A | − P a ∈ A R ( a ) is the average separation radius. R ( a ) ≡ r , Theorem 1.1 recovers(1) up to constants. We note that while ddim( A ) may be arbitrarily smallerthan ddim(Ω), it may also be larger, as A may lack points used as ball centersin coverings of Ω. However, [16] demonstrated that for all A ⊆ Ω, we haveddim( A ) ≤ Related work.
The only tangentially relevant works we found study the algorith-mic [17, 18] and game-theoretic [19] aspects of optimization problems involvingpacking different-sized items under various bin constraints. The results provedhere were early precursors to attempts at defining a useful notion of average Lip-schitz smoothness, but that line of research ended up using entirely unrelatedtechniques [20].
2. Proofs
Before proving Theorem 1.1 in its full generality, we find it instructive toprove the special case where (Ω , ρ ) is the unit ball of a d -dimensional normedspace. Any such space can be endowed with the Lebesgue measure µ such thatthe µ -volume of any r -ball is Cr d , where C depends on the norm and d only.Now if A ⊂ Ω is R -separated, then the balls B ( a, R ( a ) /
2) are all disjoint andcontained in B (0 , C d andat least C X a ∈ A ( R ( a ) / d . Combining these, we get the inequality X a ∈ A R ( a ) d ≤ d . Jensen’s inequality implies that¯ r d = | A | − X a ∈ A R ( a ) ! d ≤ | A | − X a ∈ A R ( a ) d , r d ≤ | A | − d . Solving for | A | yields the bound | A | ≤ (4 / ¯ r ) d , (3)which recovers, up to constants, the classic volumetric packing bounds (see, e.g.,[21, Lemma 5.7]) in the uniform special case R ( a ) ≡ r . The aforementionedlemma shows that d -dimensional normed spaces have ddim ≤ d log Proof of Theorem 1.1.
There is no loss of generality in normalizing all of thedistances so that diam( A ) = 1. Put N := | A | and ¯ r := N − P a ∈ A R ( a ). Wewill show that N < (5 / ¯ r ) min { ddim( A ) , ddim(Ω) } , (4)which proves the Theorem statement.To prove (4), let the Minimum Spanning Tree of A , denoted MST( A ), berooted at a point t ∈ A for which R ( t ) is minimal, and it must be that R ( t ) ≤ E be the edge-set of MST( A ), and denote the length of each edge e ∈ E by l ( e ). Further define l ( E ) = P e ∈ E l ( e ). Now assign each edge of E to theendpoint farthest from the root t ; this assigns a single edge to each point in thetree, except to the root t . Let the edge assigned to a point a ∈ A be e ( a ), andfor convenience we will say that e ( t ) is an edge of infinite length. Trivially, theedge assigned to each endpoint cannot be shorter than the distance from theendpoint to its nearest neighbor in A , so R ( a ) ≤ l ( e ( a )) for all a ∈ A . It followsthat N ¯ r = P a ∈ A R ( a ) = P a = t ∈ A R ( a ) + R ( t ) ≤ l ( E ) + 1.Now Talwar [9, Lemma 6] (see also [22, Proposition 12]) has shown that thelength of the MST on any set A ∈ Ω of N points is at most4 diam( A ) N − / min { ddim( A ) , ddim(Ω) } .
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