Plantinga-Vegter Algorithm Takes Average Polynomial Time , , and Relies on Techniques That Are Only Useful for Gaussian Mea- 3.2 Average and Smoothed Complexity Sures

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Ergur,¨ and Josue Tonelli-Cueto exchange of ideas between different approaches to continuous 2 THE PV ALGORITHM computation. Given a real smooth hypersurface in Rn described implicitly by a map f : Rn R and a region a, a n, the PV Algo- → [− ] rithm constructs a piecewise-linear approximation of the intersection of its zero set VR f with a,a n isotopic to this ( ) [− ] 1.1 Notation intersection inside a,a n. [− ] m Let m be the set of m-cubes of R . Recall that an interval roughout the paper, we will assume some familiarity with I approximation of a function F : Rm Rm′ is a map F : the basics of differential geometry and with the sphere Sn as → [ ] m such that for all J m, F J F J (c.f. [17]). a Riemannian manifold. For scalar smooth maps f : Rm R, I → Im′ ∈ I ( ) ⊆ [ ]( ) m m → We notice that if we see J as error bounds for the midpoint we will write the tangent map at x R as ∂x f : R R ∈ → m J , then F J is nothing more than error bounds for F m J . when we want to emphasize it as a linear map and as ∂f : ( ) [ ]( ) ( ( )) Assume that we have interval approximations of both f Rm Rm, x ∂f x , when we want to emphasize it as a → 7→ ( ) and its tangent map ∂f or, more generally, of hf and h′∂f for smooth function. For general smooth maps F : , we n M → N some positive maps h,h′ : R 0, . e PV Algorithm will just write ∂x F :Tx Tx as the tangent map. → ( ∞) M → N on a,a n will subdivide this region into smaller and smaller In what follows, , will denote the set of real polyno- [− ] Pn d cubes until the condition mials in n variables with degree at most d, n,d the set of + H homogeneous real polynomials in n 1 variables of degree d, Cf J : either 0 < hf I or 0 < h′∂f J , h′∂f J and and , will denote the usual norm and inner prod- ( ) [ ]( ) h [ ]( ) [ ]( )i k k h i uct in Rm as well as the Weyl norm and inner product in m is satisfied in each of the n-cubes J of the obtained subdivi- Pn,d , n m h sion of a a . In Section 4, we will be more precise on the and . Given a polynomial f , , f , will be [− ] Hn,d ∈ Pn d ∈ Hn d assumptions on our interval approximations and the functions its homogenization and ∂f the polynomial map given by its h and h′ that we will use. partial derivatives. For details about the concrete definition of each of these notions, see Section 4. Additionally, VR f and ( ) VC f will be, respectively, the real and complex zero sets of Algorithm 2.1: Subdivision routine of PV Algorithm ( ) f . Input: a 0, and f : Rn R n ∈( ∞) → We will denote by n the set of n-cubes of R and, for a with interval approximations hf and h ∂f I [ ] [ ′ ] given J n, m J will be its middle point, w J its width, and ∈I ( ) ( ) vol J = w J n its volume. Starting with the trivial subdivision := a,a n , ( ) S n {[− ] } Also, P A will denote the probability of the event A, Ex Kg x repeatedly subdivide each J into 2 cubes until the ( ) ∈ ( ) ∈S the expectation of g x when x is sampled uniformly from K condition C J holds for all J ( ) f ( ) ∈S and Eyg x the expectation ofg y with respect to a previously ( ) ( ) n specified probability distribution of y. Output: Subdivision n of a,a S⊆I [− ] Regarding complexity parameters, n will be the number of such that for all J , Cf J is true = n+d ∈S ( ) variables, d the degree bound, and N n the dimension of , . Pn d Finally, ln will denote the natural logarithm and log the log- e procedure in Algorithm 2.1 is only the subdivision rou- arithm in base 2. tine of the PV Algorithm but it dominates its complexity in the sense that the remaining computations do not add to the final cost estimates in Landau notation. Moreover, these additional computations have been implemented only for n 3. ≤ 1.2 Outline So, proceeding as in [6], we will only analyze the complexity of the subdivision routine, keeping track of the dependency In Section 2, we discuss the PV Algorithm and the n-dimensional on n.Alsoasin[6], our complexity analysis will not deal with generalization of its subdivision method that we will analyze. the precision needed for the algorithm. In Section 3, we state the main complexity results of this paper. In Section 4, we present the geometric framework of poly- 3 MAIN RESULT nomials we will work with. Following a common practice in condition-based analysis we use homogenization to get many In this section, we outline without proofs the main results of of our results. In Section 5, we introduce the condition num- this paper. In the first part, we describe our randomness as- ber along with some of its main properties. In Section 6, we sumptions for polynomials. In the second one, we give pre- present the existing results of complexity of the subdivision cise statements for our bounds on the average and smoothed method of the PV Algorithm based on local size bound func- complexity of the PV Algorithm. tions from [6] and we relate them to the local condition number. In Section 7, we rely on the bounds for the condition num- 3.1 Randomness Model ber obtained in Section 5 to derive average and smoothed com- Most of the literature on random multivariate polynomials plexity bounds under (quite) general randomness assumptions. considers polynomials with Gaussian independent coefficients Plantinga-Vegter algorithm takes average polynomial time , , and relies on techniques that are only useful for Gaussian mea- 3.2 Average and Smoothed Complexity sures. We will instead consider a general family of measures re- e following two theorems give bounds for, respectively, the lying on robust techniques coming from geometric functional average and smoothed complexity of Algorithm 2.1. In both analysis. Let us recall some basic definitions. of them c1 and c2 are, respectively, the universal constants in P1 A random variable X is called centered if EX = 0. eorems 7.2 and 7.4. P2 A random variable X is called subgaussian if there ex- Theorem 3.1. Let f , , σ > 0, and g , a do- ist a K such that for all p 1, ∈ Pn d ∈ Pn d ≥ bro random polynomial with parameters K and ρ. e expected 1 number of n-cubes in the final subdivision of Algorithm 2.1 on E X p p K√p. ( | | ) ≤ input f , a is at most ( ) Ψ 2 e smallest such K is called the 2-norm of X. n2+3n n +16n log n n ( ) n+1 P3 A random variable X satisfies the anti-concentration d 2 max 1,a 2 2 c1c2Kρ { } ( ) property with constant ρ if if the interval approximations satisfy (4.4) and (4.5) and

2 max P X u ε u R ρε. n2+5n 7n +9n log n n ( ) n+1 { (| − | ≤ ) | ∈ } ≤ d 2 max 1,a 2 2 c1c2Kρ { } ( ) e subgaussian property (P2) has other equivalent formu- if they satisfy the hypothesis of [6]. lations. We refer the interested reader to [22]. Theorem 3.2. Let f n,d , σ > 0, and g n,d a dobro Definition 3.1. A dobro random polynomial f , with ∈ P ∈ P ∈ Hn d random polynomial with parameters K and ρ . enthe expected parameters K and ρ is a polynomial number of n-cubes of the final subdivision of Algorithm 2.1 for 1 2 input qσ , a where qσ = f + σ f g is at most d / f := c X α (3.1) ( ) k k α 2 n+1 α n2+3n n +16n log n = n ( ) n+1 1 α d d 2 max 1,a 2 2 c1c2Kρ 1 + | Õ| { } ( ) σ such that the cα are independent centered subgaussian ran- if the interval approximations satisfy (4.4) and (4.5) and dom variables with Ψ2-norm K and anti-concentration prop- ≤ 2 n+1 erty with constant ρ.A dobro random polynomial f , is n2+5n 7n +9n log n n d n ( ) n+1 1 h ∈ P d 2 max 1,a 2 2 c1c2Kρ 1 + a polynomial f such that its homogenization f is so. { } ( ) σ Some dobro random polynomials of interest are the follow- if they satisfy the hypothesis of [6]. ing three. If we compare the results above with the worst-case bound N A KSS random polynomial is a dobro random polyno- of [6, eorem 4.3], which is mial such that each cα in (3.1) is Gaussian with unit nd n+1 nτ +nd log nd +9n+d log a variance. For this model we have Kρ = 1 √2π. 2O( ( ( ) ) ) / U A Weyl random polynomial is a dobro random poly- with τ being the largest bit-size of the coefficients of f , we nomial such that each cα in (3.1) have uniform distri- can see that our probabilistic bounds are exponentially bet- bution in 1, 1 . For this model we have Kρ 1. [− ] ≤ ter: they may provide an explanation of the efficiency of the E A p-random polynomial is a dobro random polyno- PV Algorithm in practice. mial whose coefficients are independent identically We note, however, that the bound in [6] and our bounds distributed random variables with the density func- cannot be directly compared. Not only because the former t p tion g t = cpe with cp being the appropriate ( ) −| | is worst-case and the laer average-case (or smoothed) but constant and p 2. ≥ because of the different underlying complexity seings: the Remark 3.2. When we are interested in integer polynomials, bound in [6] applies to integer data, ours to real data. A first dobro random polynomials may seem inadequate. One may approach to bridge this difference relies on the approximation of distributions described in Remark 3.2. But, as mentioned be inclined to consider random polynomials f n,d such τ ∈τ P there, this approach does not give completely satisfactory re- that cα is a random integer in the interval 2 , 2 , i.e., cα is [− ] a random integer of bit-size at most τ. As τ and aer we sults. A more detailed study of how the PV Algorithm behaves →∞ normalize the coefficients dividing by 2τ , this random model on random integer polynomials is desirable. converges to that of Weyl random polynomials. To have a more satisfactory understanding of random inte- 4 GEOMETRIC FRAMEWORK ger polynomials, one has to consider random variables with- ere is an extensive literature on norms of polynomials and out a continuous density function. e techniques used in this their relation to norms of gradients in , . We can use ho- Hn d note are already extended to include such random variables in mogenization to carry these results from , to , . Hn d Pn d the case of random matrices [19, 22]. We hope to pursue this To be more precise, let ϕ : Rn Sn, given by → delicate case in a more general seing (including complete in- T tersections) in future work. x 1 xT 1 + x 2. 7→ / k k p , , Felipe Cucker, Alperen A. Ergur,¨ and Josue Tonelli-Cueto

Rn is gives a diﬀeomorphism between and the upper half with ∂ϕ x f Sn √d f , by the Exclusion Lemma [1, Sn ( ) | ≤ k k of , and we have Lemma 19.22]. us taking norms and using (4.3) ﬁnishes the

h 2 d 2 proof. f ϕ x = f x 1 + x / . (4.1) ( ( )) ( )/( k k ) e claim about f˜ x is just [1, Lemma 16.6]. Using the chain rule, we can see that ( )

T Corollary 4.2. Let f . en f and ∂f are Lipschitz ∂x f d f x x n,d ∂ f h ∂ ϕ = · ( ) (4.2) ∈ P ϕ x x + 2 d 2 + 2 d 2+1 with Lipschitz constants 1 + √d and 1 + √d 1 , respectively, ( ) 1 x / − 1 x / ( ) ( − ) ( k k ) ( k k ) and for all x, f x , ∂f x 1 + xb2. c h n+1 n n where ∂y f : R R, ∂x ϕ : R Tx S = x⊥ and | ( )| k ( )k ≤ k k n → → h p ∂x f : R R are respectively the tangent maps of f , ϕ Proof. e claims about f are immediate from Proposi- → b c and f . tion 4.1. For the claims about ∂f , observe that ∂f n ∈ Pn,d 1 As it will be useful later, we note that a direct computation and that, by a direct computation,b ∂f d f . us Propo-− k k ≤ k k shows sition 4.1 completes the proof. c 2 1 2 ∂xϕ = 1 1 + x and ∂xϕ− = 1 + x . (4.3) k k / k k k k k k 4.2 Interval approximations An inner productp on , with desirable geometric prop- Hn d e Lipschitz properties above (Corollary 4.2) ensure that the erties is known as the Weyl inner product and it is given by mapping 1 d − f ,g W := fαgα J f m J + 1 + √d √nw J 1 2, 1 2 h i α 7→ ( ( )) ( ) ( ) [− / / ] α Õ is an interval approximation of f and the mapping = α = α n,d for f α fα X , g α gα X n,d . is product is b ∈ P h∈ Hh k n extended to , using f ,g := f ,g and to using J ∂f m J + 1 + √d 1 √n w J 1 2, 1 2 Í Pn d hÍ i h i Pn,d 7→ ( ( )) − ( )[− / / ] k f, g := = fi ,gi . one of ∂f . Also, leing h i i 1h i We will use to denote both the Weyl norm for polyno- c Í k k n 1 1 mials and the usual Euclidean norm in R . h x = and h′ x = ( ) f 1 + x d 1 2 ( ) d f 1 + x d 2 1 k k( k k)( − )/ k k( k k) / − 4.1 Lipschitz properties the mappings above become interval approximations hf [ ] and h ∂f of hf and h ∂f , respectively, satisfying that, for Given f n,d , let us consider the maps [ ′ ] ′ ∈ P each J n, 2 d 1 2 ∈I f : x f x f 1 + x ( − )/ 7→ ( )/ k k( k k ) dist hf J , hf m J 1 + √d √n w J 2 (4.4) ( [ ]( ) ( )( ( ))) ≤ ( )/ and b and 2 d 2 1 ∂f : x ∂f x d f 1 + x / − 7→ ( )/ k k( k k ) dist h′∂f J , h′∂f m J 1+√d 1 nw J 2. (4.5) which are just ”linearized” version of f and its derivative. e [ ]( ) ( )( ( )) ≤ − ( )/ c intuition behind this fact is that for large values of x a polyno- With these conditions on our interval approximations, we can reformulate a weaker, but easier to check, condition C J . mial map of degree d grows like x d . f′ k k ( ) Proposition 4.1. Let f k be a polynomial map. en Theorem 4.3. Let J n and assume that our interval ap- ∈ Pn,d ∈ I the map proximation satisﬁes (4.4) and (4.5). If the condition

2 d 1 2 f m J > 1 + √d √nw J x f˜ x := f x f 1 + x ( − )/ 7→ ( ) ( )/ k k( k k ) C ′ J : ( ( )) ( ) ( ) f ( ) + √  or ∂f m J > √2 1 d 1 nw J is 1 + √d -Lipschitz and, for all x, f˜ x 1 + x 2.  b ( ( )) ( − ) ( )

( ) ( ) ≤ k k is satisﬁed, then C J is true. p  f ( ) c Proof. For the Lipschitz property, it is enough to bound  Lemma 4.4. Let x Rn . en for all v,w B x , the norm of the derivative of the map by 1 + √d.Dueto(4.1), ∈ ∈ x √2( ) v,w > 0. k k/ 2 d 1 2 2 h h i f x f 1 + x ( − )/ = 1 + x f ϕ x f ( )/ k k( k k ) k k ( ( ))/k k Proof. Let W := u Rn x,u x u √2 be the p { ∈ | h i≥k kk k/ } and so the derivative equals convex cone of those vectors u whose angle x, u with x is at , , , + , fh ϕ x T ∂ f most π 4. For v w W , v w v x x w π 2, by the x + + 2 ϕ x / ∈ ≤ ≤ / ( ( )) 1 x ( ) ∂xϕ. triangle inequality. us cos v,w 0. d f 1 + x 2 k k f ≥ k k k k Now, dist x, ∂W = mindx ud x,du = x u √2 k k p ( ) {k − k | h i k kk k/ } Now, fh ϕ x pf 1,by [1, Lemma 16.6], and where the laer equals the distanced of x to a line having an ( ( )) /k k ≤ √ angle π 4 with x, which is x 2. Hence B x √2 x W = / k k/ k k/ ( ) ⊆ ∂ϕ x f ∂x ϕ ∂ϕ x f Sn ∂x ϕ and we are done. ( ) ( ) | Plantinga-Vegter algorithm takes average polynomial time , ,

Proof of Theorem 4.3. When the condition on f m J is 5.2 A fundamental proposition ( ( )) satisfied, (4.4) guarantees that 0 < hf J . Whenever the [ ]( ) e following result plays a fundamental role in our develop- condition on ∂f m J is satisfied, (4.5) and Lemmab4.4 guar- ment. Despite having been used in many occasions within var- ( ( )) antee that 0 < h′∂f J , h′∂f J . Hence C ′ J implies ious proofs, it wasn’t explicitly stated until recently. h [ ]( ) [ ]( )i f ( ) c Cf J under the given assumptions. n ( ) Lemma 5.4. Let F , and y S . en either ∈ Hn d ∈ Remark 4.5. e interval approximations in [6] are based on F y F 1 √2κ F,y Taylor expansion at the midpoint, so they are different from | ( )|/k k ≥ / ( ) ours. However, our complexity analysis also applies to the in- or terval approximations considered in [6], see §6.2 below for the ∂ n √ √ , . y F Ty S d F 1 2κ F y details. k | k/ k k ≥ / ( ) Proof. is is [3, Proposition 3.6]. 5 CONDITION NUMBER n Proposition 5.5. Let f , and x R . en either As other numerical algorithms in computational geometry, the ∈ Pn d ∈ PV Algorithm has a cost which significantly varies with inputs 1 1 f x > or ∂f x > . of the same size. One wants to explain this variation in terms | ( )| 2√2dκ f , x ( ) 2√2dκ f , x aff( ) aff( ) of geometric properties of the input. Condition numbers allow Proof.b Without loss of generality c assume that f = 1. for such an analysis. k k Let y := ϕ x , F := f h and assume that the first inequality does Definition 5.1. [2, 11] Given F , , the local condition ( ) ∈ Hn d not hold. en, by (4.1), F y 1 2√2dκ F,y 1 + x 2 . number of F at y Sn is | ( )| ≤ / ( ) k k ∈ By (4.2), (4.3) and Lemma 5.4, we get p = 2 + 2 κ F,y : F F y ∂y F T Sn d. T 2 ( ) k k/ ( ) k y k / 1 ∂x f d f x x 1 + x | ( ) k k . q √ ≤ + 2 d 2 − + 2 d 2+1 √ Given f n,d , the local affine condition number of f at x 2κ F,y 1 x / 1 x / d Rn ∈ P = h ∈ ( ) ( k k ) ( k k ) is κaff f , x : κ f , ϕ x . ( ) ( ( )) We divide by √d and use the triangle inequality to obtain 1 ∂x f f x x 5.1 What does κaff measure? k k + | ( )| k k . + 2 d 2 1 + 2 d 1 2 2 e nearer the hypersurface VR f is to having a singularity √2dκ F,y ≤ d 1 x / − 1 x ( − )/ 1 + x ( ) ( ) ( k k ) ( k k ) k k at x Rn , the smaller are the boxes drawn by the PV Algo- ∈ Using (4.1) and our initial assumption on the secondp term rithm around x. A quantity controlling how close if f to have in the sum, which we subtract, we get the desired inequality a singularity at x will therefore control the size of these boxes. since x < 1 + x 2. is is precisely what κaff f , x does. k k k k ( ) p Theorem 5.2 (Condition Number Theorem). Let x Rn 6 ADAPTIVE COMPLEXITY ANALYSIS ∈ and Σx be the set of hypersurfaces having x as a singular point. As stated in Section 2 (and as done in [6]), our complexity anal- at is, Σx := g , g x = 0, ∂xg = 0 . en for every ysis will focus on the number of subdivisions steps of the sub- { ∈ Pn d | ( ) } f , , division routine of the PV Algorithm (Algorithm 2.1). at is, ∈ Pn d f κaff f , x = dist f , Σx our cost measure will be the number of n-cubes in the final k k/ ( ) ( ) subdivision of a,a n. where the distance is induced by the Weyl norm of n,d . [− ] P We note that we do not deal with the precision needed to Proof. is follows from [1, Proposition 19.6],[2, eorem run the algorithm. is issue should be treated in the future. 4.4] and the definition of κaff. 6.1 Local size bound framework [6] eorem 5.2 provides a geometric interpretation of the local e original analysis in [6] was based on the notion of local condition number, and a corresponding ”intrinsic” complexity size bound. parameter as desired by the authors of [6, 7]. e next result will be useful in the probabilistic analyses. Definition 6.1. A local size bound for f is a function bf : Rn Rn n 0, such that for all x , Corollary 5.3. Let x R and let Px : , Σ be →[ ∞) ∈ ∈ Pn d → ⊥x the orthogonal projection onto the orthogonal complement of the b x inf vol J x J n and C J false . f ( ) ≤ { ( )| ∈ ∈I f ( ) } linear subspace Σx . en κ f , x = f Px f . aff( ) k k/k k Arguing as in [6, Proposition 4.1], one can easily get the following general bound. Proof. We have that dist f , Σx = Px f since Σx is a ( ) k k linear subspace. Hence eorem 5.2 finishes the proof. Proposition 6.2. [6] e numberof n-cubes of the final subdivision of Algorithm 2.1 on input f , a , regardless of how the We notice that the above expression should not come as a ( ) subdivision step is done, is at most surprise, since κaff is define in a way that the denominator is the norm of a vector depending linearly of f . 2a n inf b x x a,a n . ( ) / { f ( ) | ∈ [− ] } , , Felipe Cucker, Alperen A. Ergur,¨ and Josue Tonelli-Cueto

e bound above is worst-case, it considers the worst b x for which we use that f ( ) among the x a, a n. Continuous amortization developed ∈ [− ] 1 ln 1 + 22 2n 22n 3 and 1 ln 1 + 22 4n 24n 3. by Burr, Krahmer and Yap [4, 5], provides the following reﬁned / − ≤ − / − ≤ − complexity estimate [6, Proposition 5.2] which is adaptative.

Theorem 6.3. [4–6] e number of n-cubes of the final sub- e main difference betweenC f , x and κ f , x is thatC f , x division of Algorithm 2.1 on input f , a is at most ( ) ( ) ( ) ( ) is a non-linear quantity and is hard to compute, while the local n condition number κ f , x —as indicated in Corollary 5.3—is a 2 ( ) max 1, dx . linear quantity and is rather easy to compute. eorem 6.6 be- ( a,a n bf x ) ¹[− ] ( ) low and the complexity analysis in Section 7 show that the lo- Moreover, the bound is finite if and only if the algorithm termi- cal condition number κ f , x is easily amenable to the adap- aff( ) nates. tive complexity analysis techniques developed by Burr, Krah- To effectively use eorem 6.3 we need to explicit estimates mer and Yap [4, 5]. for the local size bound. 6.3 Condition-based complexity 6.2 Construction in [6] e following result expresses a local size bound in terms of the local condition number κ f , x directly, without using In [6], the authors use the following function aff( ) the construction in [6]. n 1 2 2n 2 − d ln 1 + 2 − + √n 2 f , x := min / / , Theorem 6.6. Assume that the interval approximation sat- C( ) dist x,VC f ( ( ( )) isfies (4.4) and (4.5). en 22n d 1 ln 1 + 22 4n + n 2 n − 5 2 ( − )/ / x 1 2 / dnκ f , x dist x, x ,VC g aff f p ) 7→ / ( ) (( ) ( )) where g is the polynomial ∂f X , ∂f Y , to construct a lo- is a local size bound for f . f h ( ) ( )i cal size bound. Proof. Let x Rn . As, by eorem 4.3, C J implies ∈ f′ ( ) Theorem 6.4. [6] Assume that the interval approximation C J , it is enough to compute the minimum volume of J n f ( ) ∈I satisfies the hypothesis of [6]. en containing x such that C J is false. is will still give a local f′ ( ) x 1 f , x n size function for f . 7→ /C( ) Since x J, x m J √nw J 2. Hence, by Corol- is a local size bound function for f . ∈ k − ( )k ≤ ( )/ lary 4.2 and Proposition 5.5, either Looking at the definition of f , x in [6] one can see that C( ) 1 1 measures how near is x of being a singular zero of f . is f m J 1 + √d √n w J 2 /C ( ( )) ≥ √ −( ) ( )/ is similar to 1 κ which, by eorem 5.2, measures how near 2 2dκaff f , x / aff ( ) is f of having a singular zero at x. e following result relates or b 1 these two quantities. ∂f m J 1 + √d 1 √n w J 2. n ( ( )) ≥ 2√2dκ f , x −( − ) ( )/ Theorem 6.5. Let d > 1 and f , . en, for all x R , aff ∈ Pn d ∈ ( ) 3n 2 is c means that C J is true if either f , x 2 d κ f , x . f′ ( ) C( )≤ aff( ) Proof. Note that Corollary 4.2 holds over the complex num- 2√2d 1 + √d √nκ f , x w J < 1 ( ) aff( ) ( ) bers as well. Due to this and the fact that VC f = VC f , we ( ) ( ) or have that + 2√2d 1 √d 1 nκaff f , x w J < 1. f x 1 + √d dist x, C f . b ( − ) ( ) ( ) ( ) ≤( ) ( Z ( )) Hence we get that Cf′ J is true when both conditions are sat- Now, if √2 1 + √d 1 dist y1,y2 , x, x < ∂f x , then ( ) ( b − ) (( ) ( )) k ( )k isfied and the inequality 1 + √d 2√d finishes the proof. √2 1 + √d 1 yi x < ∂f x . us, by Corollary 4.2, ( − )k − k k ( )k ≤ √2 ∂f yi ∂f x < ∂f x and so, by Lemmac 4.4, 0 , k ( )− ( )k k ( )k Using the results above, we get the following theorem ex- ∂f y1 , ∂f y2 . Hence c hibiting a condition-based complexity analysis of Algorithm 2.1. h (c) ( c)i c ∂f x √2 1 + √d 1 dist x,VC g . c ck ( )k ≤ ( − ) ( ( f )) Theorem 6.7. e number of n-cubes in the final subdivision e bound now follows from Proposition 5.5, together with of Algorithm 2.1 on input f , a is at most 3 n 1 + c 3n 2 ( ) 2 ( − )d √n 2 − d and n n n log n+9n 2 E n ≤ d max 1, a 2 / x a,a n κaff f , x 2n 1d √n 22n d 1 n { } ∈[− ] ( ) min − + , ( − ) + if the interval approximation satisfies (4.4) and (4.4), and at most + 2 2n + 2 4n ln 1 2 − 2 ln 1 2 − 2 r 2n n 3n2+2n n d max 1,a 2 Ex a,a n κaff f , x 3n 4 √n { } ∈[− ] ( ) 2 − d + , ≤ 2 if the interval approximation satisfies the hypothesis of [6]. Plantinga-Vegter algorithm takes average polynomial time , ,

Proof. is is just eorems 6.3, 6.6 and 6.5 combined with loss of generality, we will assume c1c2Kρ 1. Moreover, since , n ≥ the fact that the integral a,a n κaff f x dx is nothing more κaff is scale invariant, we can assume, again without loss of n E [− ] n ( ) than 2a x a,a n κaff f , x . generality, that c1K 1. ( ) ∈[− ] ( ∫ ( ) ) ≥ We observe that in contrast with the complexity analyses Proof of Theorem 7.1. κ f , x = f Px f by Corol- (of condition numbers closely related to κ ) in the literature aff( ) k k/k k aff lary 5.3. So, by an union bound, for all u,s > 0, (see, e.g., [2, 3, 8–11]), the bounds in eorem 6.7 depend on E n n x a,a n κaff f , x and not on maxx a,a κaff f , x . Whereas P P + P ( ( ) ) ( ) κaff f , x s f u Px f u s . (7.1) the∈[− former] has finite expectation (over f∈[−), the] laer has not. ( ( ) ≥ )≤ (k k ≥ ) (k k ≤ / ) is shows that condition-based analysis combined with adap- By eorems 7.2 and 7.4, we have tive complexity techniques such as continuous amortization n+1 may lead to substantial improvements. 2 2 uc2ρ P κaff f , x s exp 1 u c1K + . ( ( )≥ )≤ ( − /( ) ) s√n + 1 7 PROBABILISTIC ANALYSES N n+1 In this section, we prove eorems 3.1 and 3.2 stated in Sec- We set u = c1K N ln s and use s s , so we get ( ) − ≤ −( ) tion 3. p n+1 n+1 c c Kρ√N ln s 2 P , 1 2 . 7.1 Average Complexity Analysis κaff f x s 4 (n)+1 ( ( )≥ )≤ √n + 1 ! s e following theorem is the main technical result from which 1 the average complexity bound will follow. By substituting s = t n we are done. Theorem 7.1. Let f , be a dobro random polynomial ∈ Pn d with parameters K and ρ. For all x Rn and t en, Combining eorem 6.7 with the next theorem, we get the ∈ ≥ n+1 n+1 proof of eorem 3.1. c c Kρ√N ln t 2 P , n 1 2 κaff f x t 4 ( +) 1 ( ) ≥ ≤ n n + 1 1 n Theorem 7.7. Let f n,d be a dobro random polynomial ! t ∈ P ( ) with parameters K and ρ. en where c1 and c2 are, respectively,p the universal constants of e- 2 orems 7.2 and 7.4. n2+n n +3 log n +9 + E E n 2 2 ( ) n 1 f x a,a n κaff f , x d 2 c1c2Kρ e proof of eorem 7.1 relies on two basic results from ∈[− ] ( ) ≤ ( ) geometric functional analysis. where c1 andc2 are the universal constants of eorems 7.2 and 7.4. Theorem 7.2. [22, eorems 2.6.3 and 3.1.1] ere is a universal constant c1 1 with the following property. For all random Proof. By the Fubini-Tonelli theorem, ≥ vectors X = X ,..., X T with each X centered and sub- 1 N i n n ( ) E E n κ f , x = E n E κ f , x Gaussian with ψ2-norm K, and for all t c1K√N the following f x a,a aff x a,a f aff ≥ ∈[− ] ( ) ∈[− ] ( ) inequality is satisfied so it is enough to have a uniform bound for 2 2 P X t exp 1 t c1K . (k k ≥ )≤ − /( ) n = ∞ n Ef κaff f , x P κaff f , x t dt. Definition 7.3. e concentration function of a random vec- ( ) 1 ( ) ≥ Rk = P ¹ tor X is the function X ε : maxu Rk X u ε . ∈ L ( ) ∈ (k − k ≤ ) Now, by eorem 7.1, this is bounded by Theorem 7.4. [20, Corollary 1.4] ere is a universal con- n+1 + stant c2 1 with the following property. For every random vec- n 1 c1c2Kρ√N ln t 2 = ≥ ,..., T en + 4 ∞ ( ) dt. tor X X1 XN with independent random variables Xi , 1+1 n ( ) RN n n + 1 ! 1 t / and every k-dimensional linear subspace S of we have ( ) ¹ k Aer the change ofp variables t = ens the integral becomes √ P X ε k c2 max Xi ε , L k ( ) ≤ 1 i N L ( ) + + + ≤ ≤ ∞ n 1 s n 3 n 3 where P is the orthogonal projection onto S. n ns 2 e− ds = n 2 Γ , k ( ) 2 ¹0 Remark 7.5. In [15] and references therein one can find infor- where Γ is Euler’s Gamma function. Using the Stirling esti- mation about the optimal value of the absolute constant c2 in eorem 7.4. mates for it, we obtain n+2 n+2 Remark 7.6. We notice that for a dobro random polynomial + + 2 + 2 Γ n 3 n 3 n 3 f with parameters, K and ρ we have Kρ 1 [13, (1)]. Actu- √2π 4 ≥ 4 2 ≤ 2e ≤ 4 ally, the product Kρ is invariant under scaling in the following sense; for t > 0, t f is again a dobro polynomial with parame- and N 2d n. Combining all these inequalities, we obtain ≤ ( ) ters tK and ρ t. Hence, for the sake of simplicity and without the desired upper bound. / , , Felipe Cucker, Alperen A. Ergur,¨ and Josue Tonelli-Cueto

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