On the Discrepancy Normed Space of Event Sequences for Threshold-based Sampling

On the Discrepancy Normed Space of Ev en t Sequences for Threshold-based Sampling Bernhard A. Moser Soft ware Competence Center Hagen b erg, Austria Email: b ernhard.moser@scc h.at No vem b er 10, 2021 Abstract Recalling recen t results on the characterization of threshold-based sampling as quasi-isometric mapping, mathematical implications on the metric and topological structure of the space of even t sequences are deriv ed. In this con text, the space of ev ent se quences is extended to a normed space equipp ed with Hermann W eyl’s discrepancy measure. Sequences of finite discrepancy norm are c haracterized by a Jordan de- comp osition property . Its dual norm turns out to b e the norm of total v ariation. As a by-product a measure for the lack of monotonicit y of sequences is obtained. A further result refers to an inequalit y betw een the discrepancy norm and total v ariation whic h resem bles Heisen b ergs uncertain ty relation. Keyw ords: Quasi Isometry , Discrepancy Measure, Alexiewicz Norm, T otal V ariation, Dual Norm, Jordan Decomp osition 1 Motiv ation This pap er starts by recalling a recen t result for the understanding of threshold- based sampling sc hemes as quasi-isometric mapping [1]. In this con text a threshold-based samplin g sc heme is understo o d as a mapping from the space of sampled signals to the space of resulting event se quenc es of “up” and “do wn” even ts that preserves the notion of “closeness” or synonymously “similarit y”. The “up” and “down” even ts are triggered by the sampling pro cess. Usually these even ts are represented by +1 and − 1, resp ectively . T o b e precise, preserving the top ology is not p ossible in the strict sense (see e.g. [2]). This effect is an immediate consequence of the all-or-nothing law of threshold-based sampling. Either there is a triggering sampling even t at a certain time or not. T ak e for example signals b elo w threshold. Such signals cannot be distinguished from the samples, b ecause there are none. So pre- serving the metric as e.g. the notion of closeness can only b e satisfied in a 1 relaxed fashion, namely as quasi-isometry . As a consequence, w e single out metrics b eing compatible with the quasi-isometry constrain t. As p ointed out in [1], this analysis leads to the class of metrics for which a sequence of alternating “up” (mo deled b y 1) and “down” (mo deled b y − 1) even ts is considered to be close to the zero sequence that con tains no even t at all. One metric that fulfils this condition is due to Hermann W eyl, namely the so-called discrepancy measure (see, [3 – 5]). This measure was in tro duced o ver 100 y ears ago in the con text of ev aluating the quality of pseudo-random n umbers. In a vector space this measure leads to a norm, the discrepancy norm k . k D . This norm distinguishes itself from the familiar Euclidean or another L p norm by its asymmetric shap e of its unit ball. This asymmetry is due to the fact that the norm ev okes in general differen t lengths after rearranging the order of even ts in a sequence. There is an instructive inter- pretation of the discrepancy . Consider a w alker along a line, who makes a step ahead if the even t is “up” and a step backw ards, if the even t is “down”. The discrepancy is the range of the walk. As shown in [6], t ypical metrics in this context such as the v an Rossum [7] or the Victor-Purpura metric [8] do not satisfy this condition. As a conse- quence arbitrary small deviations can cause disruptiv e effects in the input- output b ehavior when relying on similarity measures based on such metrics. In this pap er we fo cus on mathematical implications on the topological structure of the space of even t sequences when underlying W eyl’s discrep- ancy norm k . k D . As first result, we pro vide a characterization of those even t sequences that are finite in this metric in a w a y that resembles the Jordan decomp osition law of functions of total v ariation, see Section 3.1. This result indicates a close relationship b etw een the discrepancy norm, k . k D , and the semi-norm of total v ariation, k . k BV . In analogy to L p spaces w e denote the space of even t sequences that are b ounded with resp ect to k . k D b y L D . In Section 4 w e study the dual space L ∗ D of L D . As second result we identify L ∗ D as the space of functions of total v ariation. As measure of oscillation k . k BV b eha ves inv erse prop ortional to k . k D . If the range of a walk consisting of +1 and − 1 steps of length n ∈ N is small then there is muc h oscillation. F or example, for a sequence of alternating signs, +1 , − 1 , +1 . . . , the range is minimal and the oscillation is maximal. On the other hand, little oscillation means that there is a predominan t direction of the walk and therefore a larger range. This recipro cal relation is topic of Section 5, which leads to the inequality ( x i ∈ {− 1 , 1 } , x = ( x 1 , . . . , x n ) not constan t) n ≤ k x k D · k x k BV , whic h resembles Heisen b erg’s uncertaint y relation in its form. On the left hand side there is a constant as lo w er b ound and on the righ t hand side there is a product of tw o measures that represen t dual concepts. F or the Heisen b erg inequalit y these dual concepts are time and frequency . In our 2 case, the dual concepts refer to oscillations in terms of total v ariation and range of the corresp onding walk. Before, we start with a section on preliminaries (Section 2) by intro- ducing and recalling the notion of quasi-isometry (Subsection 2.1), W eyl’s discrepancy (Subsection 2.2) and its relation to quasi-isometry in the context of threshold-based sampling (Subsection 2.3). 2 Preliminaries 2.1 Mathematics of Distances First of all, let us fix some notation. 1 I denotes the indicator function of the set I , i.e., 1 I ( t ) = 1 if t ∈ I and 1 I ( t ) = 0 otherwise. k . k ∞ denotes the uniform norm, i.e., k f − g k ∞ = sup t ∈ X | f ( t ) − g ( t ) | , where X is the domain of f and g . If M is a discrete set then | M | denotes its num ber of elements. If I is an in terv al, then | I | denotes its length. I denotes the family of real in terv als. In this section we recall basic notions related to distances suc h as semi- metric, isometry and quasi-isometry , see e .g., [9]. Let X b e a set. A pseudo-metric d : X × X → [0 , ∞ ) is characterized b y a) d ( x, x ) = 0 for all x ∈ X , b) d ( x, y ) = d ( y , x ) for all x, y ∈ X and c) the triangle inequality d ( x, z ) ≤ d ( x, y ) + d ( y , z ) for all x, y , z ∈ X . d is a metric if, in addition to a) the stronger condition a’) d ( x, y ) = 0 if and only if x = y , is satisfied. The semi-metric ˜ d is called e quivalent to d , in symbols d ∼ ˜ d , if and only if there are constants A 1 , A 2 > 0 such that A 1 d ( x, y ) ≤ ˜ d ( x, y ) ≤ A 2 d ( x, y ) (1) for all x , y of the universe of discourse. A map Φ : X → Y b etw een a metric space ( X , d X ) and another metric space ( Y , d Y ) is called isometry if this mapping is distance preserving, i.e., for any x 1 , x 2 ∈ X w e hav e d X ( x 1 , x 2 ) = d Y (Φ( x 1 ) , Φ( x 2 )). The concept of quasi-isometry relaxes the notion of isometry b y imposing only a coarse Lipschitz con tin uity and a coarse surjectiv e prop erty of the mapping. Φ is called a quasi-isometry from ( X , d X ) to ( Y , d Y ) if there exist constan ts A ≥ 1, B ≥ 0, and C ≥ 0 suc h that the following tw o prop erties hold: i) F or ev ery t wo elements x 1 , x 2 ∈ X , the distance b et ween their images is, up to the additiv e constan t B , within a factor of A of their original distance. This means, there are constants A and B such that ∀ x 1 , x 2 ∈ X 1 A d X ( x 1 , x 2 ) − B ≤ d Y (Φ( x 1 ) , Φ( x 2 )) ≤ A d X ( x 1 , x 2 ) + B . (2) ii) Every element of Y is within the constant distance C of an image p oin t, 3 i.e., ∀ y ∈ Y : ∃ x ∈ X : d Y ( y , Φ( x )) ≤ C . (3) Note that for B = 0 the condition (2) reads as Lipschitz contin uity condition of the op erator Φ. This means that (2) can b e interpreted as a relaxed bi-Lipschitz condition. The tw o metric spaces ( X , d X ) and ( Y , d Y ) are called quasi-isometric if there exists a quasi-isometry Q from ( X , d X ) to ( Y , d Y ). In this pap er, the total v ariation k . k BV pla ys a cen tral role. It is a measure for the amoun t of oscillations and is defined by k f k BV := sup x 1 0 | λ x ∈ P } ) − 1 . 4 As sho wn in [5] the norm induced by SOD on the h yp ercub e yields Hermann W eyl’s discrepancy . In [3] W eyl introduces a concept of discrepancy in the context of pseudo- randomness of sequences of num b ers from the unit interv al. W eyl’s discrep- ancy concept leads to the definition k x k D = sup n 1 ,n 2 ∈ Z : n 1 ≤ n 2 , | n 2 X i = n 1 x i | , (5) whic h induces a norm on the n -dimensional real vector space [5]. Applica- tions of the norm (5) can b e found in pattern recognition [10], print insp ec- tion in the con text of pixel classification [11], template matching and regis- tration [12]. In con trast to p -norms k . k p , k x k p = ( P i | x i | p ) (1 /p ) , the norm k . k D strongly dep ends on the sign and also the ordering of the entries, as illustrated by the examples k ( − 1 , 1 , − 1 , 1) k D = 1 and k ( − 1 , − 1 , 1 , 1) k D = 2. Generally , x = ( x i ) i with x i ≥ 0 en tails k x k D = k x k 1 , and x = (( − 1) i ) i the equalit y k x k D = k x k ∞ , resp ectiv ely , indicating that the more there are alternating signs of consecutive entries, the low er is the v alue of the discrepancy norm. Observ e that k x k ∞ ≤ k x k D ≤ k x k 1 , hence, due to Ho elder’s inequality n − 1 /p k x k p ≤ k x k D ≤ n 1 − 1 /p k x k p . F or conv enience let us consider a sequence ( x i ) i with i ∈ I n , x i = 0 for i / ∈ I n , and denote by ∆ x ( k ) = k ( x i + k − x i ) i k D the misalignement function of x with respect to k . k D . Then w e hav e the follo wing properties [12]: (P1) k ( x i ) i ∈ I n k D induces a norm on R n . (P2) ∆ x (0) = 0 for all summable real sequences x . (P3) k ( x i ) i ∈ I n k D = max { 0 , max k ∈ I n P k i =1 x i } − min { 0 , min k ∈ I n P k i =1 x i } (P4) Lipsc hitz prop erty: ∆ x ( k ) ≤ | k | · L , where L = max i x i − min i x i and k ∈ Z . (P5) ∆ x ( k ) = ∆ x ( − k ) for x = ( x i ) i with x i ≥ 0 and k ∈ Z . (P6) F or x = ( x i ) i with x i ≥ 0 the function ∆ x ( . ) is monotonically increas- ing on N ∪ { 0 } . Equation (P3) allo ws us to compute the discrepancy of a sequence of length n with O ( n ) op erations instead of O ( n 2 ) num b er of op erations resulting from the original Definition (5). Esp ecially the monotonicity (P6) as well as the Lipsc hitz prop ert y (P4) are in teresting properties for applications in the field of signal analysis. It is instructive to point out that the Lipsc hitz constan t in (P4) do es not dep end on frequencies or other c haracteristics of the sequence x . Prop erties (P4), (P5) and (P6) are illustrated in the Figures 2(a) and 2(b) whic h demonstrate the b ehavior of the misalignment 5 function of a sequence of all-or-none ev en ts. While Figure 2(a) sho ws t ypical lo cal minima of the misalignment function with resp ect to the Euclidean norm, Figure 2(b) visualizes the symmetry prop erty (P5), the monotonicity prop ert y (P6) and the boundedness of its slope due to the Lipsc hitz prop erty (P4) of the corresponding misalignmen t function induced b y the discrepancy norm. (a) (b) Figure 2: Figure (a) shows a sequence of all-or-none ev ents. Figure (b) de- picts its misalignmen t function with resp ect to the Euclidean norm (dashed line) and with resp ect to the discrepancy norm (solid line). Note that the solid line is monotonic according to (P6) . T o obtain a clear in terpretation of the discrepancy , let’s think of a w alker who mov es up or down along a line at each time step according to the sequence ( x 1 , . . . , x n ) ∈ {− 1 , 1 } n . What is the range of this mov ement? Consider the pair of v ariables ( t, d ) for time and distance. The walk can b e represen ted b y the graph γ = (0 , 0) T , (1 , x 1 ) T , . . . , n X i =1 (1 , x i ) T ! in ( N 0 × Z ) n +1 . The diameter (range) of γ w.r.t. the second v ariable, i.e., in the direction of (0 , 1) T , is giv en by max 1 ≤ n 1 ,n 2 ≤ n      n 2 X i = n 1  (1 , x i ) T , (0 , 1) T       = max 1 ≤ i ≤ n i X j =0 x j − min 1 ≤ i ≤ n i X j =0 x j = k x k D (6) 6 where h ., . i denotes the usual inner pro duct, and x 0 = 0. Equation (6) tells us that the discrepancy can b e interpreted as range. It is interesting to note that this interpretation was the key to solve the problem of computing the distribution of the range of a random w alk [13]. A problem that remained unsolv ed for more than 50 years after it was s tated by F eller in 1951 [14]. 2.3 Threshold-Based Sampling as Quasi-Isometry [1] provides a framework for constructing metrics in the input and the out- put space of a threshold-based sampling scheme Φ θ suc h that Φ θ b ecomes a quasi-isometry with constan ts A θ and B θ , according to (2). The construc- tion relies on W eyl’s discrepancy norm. The metrics can b e constructed in a wa y that lim θ → 0 A θ = 1 and lim θ → 0 B θ = 0 (see Theorem 6.1 of [1]). This means that these metrics are asymptotically isometric for ev er decreasing thresholds. F or example, for Send-on-Delta (SOD) and In tegrate-and-Fire (IF) we obtain A θ = 1 and B θ = 4 θ . In b oth cases we obtain k . k D as metric in the output space, that is the space of ev en t sequences. In the input space, in the former case (SOD) we get the semi-norm of the range and for the latter (IF) we obtain as metric an in tegral v ersion of the discrepancy norm for in tegrable functions. F urther analysis shows that the c hoice of the discrepancy measure or some quasi-isometric v arian t of it is even necessary in order to turn Φ θ in to a quasi-isometry . This sp ecial role of the discrepancy measure in the context of threshold- based sampling strongly motiv ates to inv estigate the space of even t se- quences based on the discrepancy measure as metric in more detail. 3 Conception of the Space of Ev en t Sequences as Metric Space T aking up the results ab out quasi-isometry of Subsection 2.3, w e come up with the follo wing p ostulates for the space of ev ent sequences for threshold- based sampling. Basically , an even t sequence is a function in time that is zero except at discrete time p oin ts of triggered even ts. In the case of bipolar even ts we therefore hav e functions of the form η : [0 , ∞ ) → {− 1 , 0 , 1 } . As the even ts are triggered by the sampling scheme, the even ts are sparse, that is there are no accum ulation points of ev ents. Putting in other words, for any finite time in terv al [ a, b ] there are only a finite num ber of ev ents inside this in terv al. No w, let us extend this space to the vector space of functions η : [0 , ∞ ) → Z and equip this space with the discrepancy norm k . k D . Note that an even t 7 sequence can synon ymously b e represented by its sequence of ev ents ( t k , η k ) k ( η k := η ( t k )) which justifies the te rm “sequence” in this context. Therefore, the discrepancy norm of an even t sequence, k η k D is well defined by referring to the sequence, i.e., k η k D := k ( η k ) k k D = sup [ a,b ]     Z b a η dc     = sup [ a,b ]      b X k = a η ( t k )      , (7) where the last line of (7) represen ts the sum as in tegral w.r.t. the counting measure c . Let us denote this normed space of ev ent sequences η : [0 , ∞ ) → Z of finite discrepancy , k η k D < ∞ , by ( E D , k . k D ) . (8) Analogously , referring to the input s pace of signals we can equip the the space of locally in tegrable functions, P ( R ), with the discrepancy in its in tegral v ersion k f k D ,λ := sup [ a,b ]     Z b a f dλ     w.r.t. the Lebesgue measure λ . Let us denote L D := { f ∈ P ( R ) | k f k D ,λ < ∞} . (9) W e refer to the corresp onding normed space b y ( L D , k . k D ,λ ) . (10) Next, w e present the results of this pap er. First w e provide a character- ization of ev en t sequences and functions of finite discrepancy . 3.1 Jordan Decomp osition of Finite Discrepancy Sequences and F unctions The following Lemma 3.1 sho ws that k f k D ,λ can b e represented as range of v alues assumed by the function Γ f ( x ) := lim inf n ∈ Z ,n 0 implies that for all k ∈ Z there is a natural num b er N k ∈ N such that for all n ≥ N k there holds inf m ≥ n R k − m f dλ ≥ ε/ 2. Hence, there is a sequence of increasing n umbers ( k n ) n , lim n k n = ∞ , such that R k n − k n +1 f dλ ≥ ε/ 2. Con- sequen tly , we obtain sup a,b | R b a f dλ | ≥ R k 1 − k n +1 f dλ ≥ n ε/ 2 which con tradicts k f k D ,λ < ∞ . Hence, inf x Γ f ( x ) ≤ 0 . (13) Analogously , w e obtain sup x Γ f ( x ) ≥ 0 . (14) F urther, note that (12) yields Z b a f dµ = lim inf n →∞ Z b − n f dλ − lim inf n →∞ Z a − n f dλ (15) for a < b . T aking (13), (14) and (15) together prov es Lemma 3.1.  F or an example take f ( t ) = sin( t ) on R . This function has finite discrepancy , namely k f k D = R π 0 sin( t ) dt = 2. Note that in general lo cally in tegrable p erio dic functions ha ve finite discrepancy . In analogy to (11) we define γ η ( k ) := lim inf n →∞ k X j = − n η j . and obtain an iden tity in analogy to Lemma 3.1, i.e., k ( η k ) k k D = sup k ∈ Z γ η ( k ) − inf k ∈ Z γ η ( k ) , (16) where ( η k ) k ∈ R Z . It is a well kno wn result, the so-called Jordan decomp osition law, that functions f of b ounded v ariation can b e characterized as difference of mono- tonic functions h 1 and h 2 , f = h 2 − h 1 . F or functions of b ounded discrepancy w e obtain an analogous c haracterization. 9 Theorem 3.2 ( Jordan Decomp osition of Bounded Discrepancy , λ - V ersion ) L et f ∈ P ([ a, b ]) , a < b . Then k f k D ,λ ≤ r < ∞ if and only if ther e ar e non-de cr e asing lo c al ly absolutely c ontinuous functions h 1 , h 2 such that k h 2 − h 1 k ∞ ≤ r / 2 and f = ˙ h 2 − ˙ h 1 almost everywher e. F or the pro of w e split f = f + − f − in to its non-negativ e and non- p ositiv e part f + = max { f , 0 } , f − = min { f , 0 } . F or k f k D ,λ = 0 w e choose h 1 = h 2 = 0. F urther on, assume that k f k D ,λ > 0. Due to the compactness of [ a, b ] there is an interv al [ a ∗ , b ∗ ] ⊆ [ a, b ] suc h that r := k f k D ,λ = | R b ∗ a ∗ f dλ | . Due to the intermediate v alue theorem there is a real c ∗ ∈ [ a ∗ , b ∗ ] such that | R x a ∗ f dλ | = | R b ∗ c ∗ f dλ | = r / 2. Let us define h 2 ( x ) := Z x c ∗ f + dλ, h 1 ( x ) := − Z x c ∗ f − dλ. Consider the in terv als [ a k , b k ], k ∈ Z , at which | R b k a k f dλ | assumes its max- im um, that is | R b k a k f dλ | = k f k D ,λ = r . Note that h 1 and h 2 are non- decreasing and almost everywhere differentiable with f = ˙ h 2 − ˙ h 1 . F urther, note that R b k a k f dλ ∈ {− r, r } is an alternating sequence which implies that | h 2 ( x ) − h 1 ( x ) | = | R x c ∗ f dλ | ≤ r / 2 for all x , hence k h 2 − h 1 k ∞ ≤ r / 2. On the other hand, let us suppose that k h 2 − h 1 k ∞ ≤ r / 2 where h 1 , h 2 are absolutely contin uous functions satisfying f = ˙ h 2 − ˙ h 1 with ˙ h 2 , ˙ h 1 ≥ 0 a.e.. Then Lemma 3.1 entails k f k D ,λ (17) = sup x Γ f ( x ) − inf x Γ f ( x ) = sup x lim inf n →∞ Z x − n ˙ h 2 ( x ) − ˙ h 1 ( x ) dλ − inf x lim inf n →∞ Z x − n ˙ h 2 ( x ) − ˙ h 1 ( x ) dλ = sup x ( h 2 ( x ) − h 1 ( x )) − lim inf n →∞ ( h 2 ( − n ) − h 1 ( − n )) − inf x ( h 2 ( x ) − h 1 ( x )) + lim inf n →∞ ( h 2 ( − n ) − h 1 ( − n )) . Since k h 2 − h 1 k ∞ < ∞ implies that lim inf n ( h 2 ( − n ) − h 1 ( − n )) exists, that is lim inf n ( h 2 ( − n ) − h 1 ( − n )) = ρ ∈ R , Equation (17) finally implies k f k D ,λ = sup x ( h 2 ( x ) − h 1 ( x )) − inf x ( h 2 ( x ) − h 1 ( x )) ≤ 2 r 2 < ∞ , whic h ends the pro of.  In an analogous wa y w e obtain a Jordan decomp osition representation for the discrete v ersion. 10 Theorem 3.3 ( Jordan Decomp osition of Bounded Discrepancy , Dis- crete V ersion ) L et η = ( η k ) k ∈ N ∈ R N . Then k η k D = r < ∞ if and only if ther e ar e non-de cr e asing se quenc es χ 1 = ( χ 1 ( k )) k , χ 2 = ( χ 2 ( k )) k such that k χ 2 − χ 1 k ∞ ≤ r / 2 and η ( k ) = ( χ 2 ( k ) − χ 2 ( k − 1)) − ( χ 1 ( k ) − χ 1 ( k − 1)) . Assume that k η k D = r < ∞ . W e set χ ( α ) 2 ( k ) := k X i =1 max { 0 , η i } − α, (18) χ 1 ( k ) := − k X i =1 min { 0 , η i } , where α in the first line in (18) is defined b y α := 1 2  max k ∈ Z { χ (0) 2 ( k ) − χ 1 ( k ) } − min k ∈ Z { χ (0) 2 ( k ) − χ 1 ( k ) }  . F or con venience w e define χ 2 (0) := − α and χ 1 (0) := 0. Note that χ 1 and χ 2 are non-decreasing. F urther, we c heck that ( χ 2 ( k ) − χ 2 ( k − 1)) − ( χ 1 ( k ) − χ 1 ( k − 1)) = η k and that | χ 2 ( k ) − χ 1 ( k ) | ≤ r / 2 . The other direction of the proof follows. Supp ose η ( k ) = ( χ 2 ( k ) − χ 2 ( k − 1)) − ( χ 1 ( k ) − χ 1 ( k − 1)), k χ 2 − χ 1 k ∞ ≤ r / 2 and consider the range repre- sen tation of the discrepancy k η k D = max k ∈ N { 0 , k X j =1 ( χ 2 ( j ) − χ 2 ( j − 1)) − ( χ 1 ( j ) − χ 1 ( j − 1)) } − min k ∈ N { 0 , k X j =1 ( χ 2 ( j ) − χ 2 ( j − 1)) − ( χ 1 ( j ) − χ 1 ( j − 1)) } = max k ∈ N { 0 , χ 2 ( k ) − χ 1 ( k ) } − min k ∈ N { 0 , χ 2 ( k ) − χ 1 ( k ) } ≤ r . 11  As a corollary we obtain the result that a b ounded discrepancy func- tion can also b e characterized by a differentiable function whose range is b ounded. It turns out that the Leb esgue measure of this range equals the discrepancy . Corollary 3.4 ( Discrepancy as Range, Second V ersion ) L et f ∈ P ( R ) . Then k f k D ,λ = r < ∞ if and only if ther e is a uniquely determine d lo c al ly absolutely c ontinuous function g such that g ( R ) = [0 , r ] and f = ˙ g almost everywher e. Supp ose k f k D ,λ = r < ∞ . Let us in tro duce g ( x ) = − c + lim inf n →∞ Z x − n f dλ (19) where c := inf x lim inf n →∞ R x − n f dλ . Due to the fundamen tal theorem of Leb esgue integral calculus g is lo cally absolutely contin uous and differen- tiable almost everywhere. Equation (19) implies inf x g ( x ) = 0 by construc- tion. Now, consider sup x g ( x ) = − inf x lim inf n →∞ Z x − n f dλ + sup x lim inf n →∞ Z x − n f dλ whic h by Lemma 3.1 yields sup x g ( x ) = k f k D ,λ = r . The iden tity f = ˙ g almost everywhere follo ws from construction (19). No w, consider an absolutely con tinuous function g with g ( R ) = [0 , r ], i.e., inf x g ( x ) = 0 and sup x g ( x ) = r ≥ 0. Then, sup x lim inf n →∞ Z x − n ˙ gdλ = sup x lim inf n →∞ ( g ( x ) − g ( − n )) = sup x g ( x ) − lim sup n →∞ g ( − n ) and, analogously , inf x lim inf n →∞ Z x − n ˙ gdλ = inf x g ( x ) − lim sup n →∞ g ( − n ) . F rom this and Lemma 3.1 we obtain k ˙ g k D = r < ∞ . The uniqueness follows from the fundamental theorem of Leb esgue in tegral calculus and the fact that the in tegration constant is determined by the restriction inf x g ( x ) = 0.  12 4 The Dual Space E ∗ D One of the cen tral questions in functional analysis is the characterization of the dual space V ∗ of a giv en vector space V . V ∗ consists of all linear functionals L : V → R , together with the v ector space structure of p oint wise addition and scalar m ultiplication b y constants. If the vector space ( V , k . k ) is equipp ed with a norm k . k , the question arouses ab out the dual norm k . k ∗ in V ∗ , which is induced b y k L k ∗ := sup {| L ( x ) | | k x k ≤ 1 } . (20) Note that k L k ∗ exists if the linear functional L is b ounded w.r.t the norm k . k , i.e., there is a constant M > 0 such that | L ( x ) | ≤ M · k x k for all x ∈ V with k x k . In this section w e will determine the dual space E ∗ D and the corresponding dual norm (20). First of all, consider a linear functional L : E D → R , an ev ent sequence η ∈ E D and an in terv al [ a, b ]. Note that the subset of ev en ts of η contained in [ a, b ] defines also an ev ent sequence. W e denote this even t sequence by η | [ a,b ] :=  η ( t ) . . . t ∈ [ a, b ] 0 . . . else . There are only finitely many even ts in [ a, b ], say at t i j ∈ [ a, b ]. F or con ve- nience we write ( η t i j ) j := η | { t i j } . So, η t i j ∈ E D denotes that singleton ev en t sequence that is zero ev erywhere except at t i j , where the even t is giv en by η ( t i j ). F or conv enience, let us write f L ( t i j ) := L ( η t i j ) ∈ R . The following Lemma 4.1 is a direct consequence of the linearit y of L . Lemma 4.1 ( Linear F unctionals on E ) L is a line ar functional on E if and only if ther e is a unique function f L : [0 , ∞ ) → R , such that for al l [ a, b ] ⊆ [0 , ∞ ) and al l event se quenc es η ∈ E ther e holds L ( η | [ a,b ] ) = X t i j ∈ [ a,b ] f L ( t i j ) · η t i j . Next we characterize those linear functionals which are b ounded w.r.t k . k D . Theorem 4.2 ( Bounded Linear F unctionals on E D ) L ∈ E ∗ is b ounde d w.r.t the discr ep ancy norm k . k D if and only if f L has b ounde d variation, i.e., k f L k BV < ∞ . Supp ose that L is b ounded. Indirectly , supp ose that k f L k BV = ∞ . Then there is a sequence of partitions P k = { t ( k ) 1 , . . . , t ( k ) n k } such that sup k →∞ n k X j =1 | f L ( t ( k ) j +1 ) − f L ( t ( k ) j ) | = ∞ . 13 This means that either (the summation is taken ov er all defined indexes) sup k →∞ X j | f L ( t ( k ) 2 ∗ j ) − f L ( t ( k ) 2 j − 1 ) | = ∞ (21) or sup k →∞ X j | f L ( t ( k ) 2 ∗ j +1 ) − f L ( t ( k ) 2 j ) | = ∞ . (22) Note that | f ( t i +1 ) − f ( t i ) | = f ( t i ) η ( t i ) + f ( t i +1 ) η ( t i +1 ) , (23) where ( η ( t i ) , η ( t i +1 )) := (1 , − 1) if f ( t i ) ≥ f ( t i +1 ) and ( η ( t i ) , η ( t i +1 )) := ( − 1 , 1) if f ( t i ) < f ( t i +1 ). (23) together with (21), (22) means that there is an even t sequences η ( k ) suc h that the corresp onding sequence of summations ψ k := P j f L ( t j ) η ( k ) ( t j ) = L ( η ( k ) ) is unbounded, which contradicts the assumption that L is b ounded. Hence, k f L k BV < ∞ . No w, supp ose that k f L k BV < ∞ . Consider an even t sequence η with k η k D ≤ 1. This means that the corresp onding sequence of even ts ( η ( t i )) i is alternating in sign. Consequen tly , we obtain | L ( η | [ a,b ] ) | = | X t i j ∈ [ a,b ] f L ( t i j ) · η t i j | ≤ | f L ( t 1 ) − f L ( t 2 ) | + . . . + | f L ( t n − 1 ) − f L ( t n ) | ≤ k f L k BV , (24) for an y interv al [ a, b ] and any c hoice of partitions ( t 1 , . . . , t n ). Hence, L is b ounded.  (24) implies the follo wing Proposition (4.3). Prop osition 4.3 ( Dual Discrepancy Norm ) L et f b e of b ounde d variation, k f k BV < ∞ . Then 1 2 k f k BV ≤ k f k ∗ D ≤ k f k BV , wher e k f k ∗ D = sup η ∈ E D ,η 6 = 0 | L f ( η ) | k η k D and L f is the c orr esp onding line ar functional induc e d by f . 14 Note that if f is monotonic, w e get k f k ∗ D = k f k BV , and if f is p erio dically oscillating such as f ( t ) = sin( t ) on [ a, b ] w e obtain a lo w measure. This means that µ mon ( f ) := k f k ∗ D k f k BV (25) measures to which extent a non-constant function f , i.e., k f k BV > 0, is monotonic. Note that µ mon ( f ) can b e computed in O ( n ) if f is discrete giv en by n v alues. This can b e achiev ed by identifying local extremal p oints of f as p oints of even ts (“up”-even t for lo cal m axim um and “down”-ev en t). Compared to monotonicity measures [15] based on rearranging the ordering in order to ac hieve monotonicit y whic h is of O ( n log ( n )), our measure (25) distinguishes b y its lo w computational complexit y of O ( n ). A detailed study of the features of this monotonicit y measure will b e p ostp oned to future researc h. Is there an equiv alent discrepancy measure that yields the total v ariation as its dual norm? Y es! W e just need a sligh t mo dification of the discrepancy norm, the Alexiewicz norm [16]: k ( x 1 , . . . , x n ) k A := max k {| k X i =1 x i |} (26) Note that k x k D / 2 ≤ k x k A ≤ k x k D . Note that − 1 , 2 , − 2 . . . is a sequence with Alexiewicz norm 1. Let t i mark lo cations of lo cal minimum or maximum, whic h are alternating. Let t 0 denote the first local extremum. If t 0 marks a local minimum, then w e set η ( t 0 ) := − 1, if it is a lo cal maxim um w e set η ( t 0 ) := +1. Then we pro ceed b y consecutively assigning ± 2 alternating in sign at the p ositions t i of lo cal extrema. By this w e obtain X i f ( t i ) η ( t i ) = X i | f ( t i +1 − f ( t i )) | , hence k L f k ∗ A = sup η ∈ E A ,η 6 = 0 | L f ( η ) | k η k A = k f k BV , (27) where L f denotes the linear functional induced b y f . 5 A Heisen b erg-t yp e Inequalit y b et w een Discrep- ancy and T otal V ariation Consider x ∈ {− 1 , 1 } n , whic h is not constant, that is k x k BV > 0. First, let us c haracterizes sequences x of minimal total v ariation k x k BV = 2. k x k BV = 2 15 is the case if and only if there is one c hange in sign, hence, up to choosing the initial sign, w e ha ve x = (1 , . . . , 1 | {z } k , − 1 , . . . , − 1 | {z } n − k ) . Note that n 2 ≤ max { k , n − k } = k x k D , hence n ≤ k x k D · k x k BV . F or an arbitrary num b er S of changes in the sign w e ha ve k x k BV = 2 · S and n S ≤ max ( k 1 , . . . , k S : X i k i = n, k i ∈ N ) = k x k D . Consequen tly , we obtain n ≤ n S S · 2 ≤ k x k D · k x k BV . This result also applies to x ∈ {− 1 , 0 , 1 } n . T o see this, first cancel all zeros from x whic h yields ˆ x and note that k x k 1 = k ˆ x k 1 ≤ k ˆ x k D · k x k BV ≤ k x k D · k x k BV . (28) Finally , (28) implies the Heisen b erg-t yp e inequality b etw een discrepancy and total v ariation k x k 1 ≤ k x k D k x k BV (29) for even t sequences x ∈ {− 1 , 0 , 1 } n , n ∈ N and k x k BV > 0. This inequalit y is sharp as x = (1 , . . . , 1 , 0 , . . . , 0) or x = (1 , . . . , 1 , − 1 , . . . , − 1) induce equality . By in terpreting k x k BV in (29) as measure of oscillation and k x k D as me asure of the dominance of monolithic blo cks, we recognize the similarit y to Heisen b erg’s inequalit y relation. 6 Conclusion In this article w e in vestigated the space of even t sequences as normed s pace equipp ed with W eyl’s discrepancy norm, which distinguishes b y its property to turn threshold-based sampling in to a quasi-isometry mapping. As result w e found v arious characterizations and interpretations of this norm as for example a Jordan-like decomp osition law. W e also inv estigated its relation- ship to total v ariation and found a Heisenberg-type inequality . The ratio of the dual discrepancy norm and total v ariation turns out to b e a measure of monotonicit y , which will b e in vestigated in more detail in the future. 16 References [1] B. A. Moser, “Similarity reco v ery from threshold-based sampling under general conditions,” IEEE T r ans. Signal Pr o c essing , v ol. 65, no. 17, pp. 4645–4654, 2017. [2] B. A. Moser, “On preserving metric properties of in tegrate-and-fire sampling,” in 2016 Se c ond International Confer enc e on Event-b ase d Contr ol, Communic ation, and Signal Pr o c essing (EBCCSP) , pp. 1–7, June 2016. [3] H. W eyl, “ ¨ Ub er die Gleic hv erteilung von Zahlen mod. Eins,” Mathema- tische A nnalen , v ol. 77, pp. 313–352, 1916. [4] C. Doerr, M. Gnewuch, and M. W ahlstr¨ om, Calculation of Discr ep ancy Me asur es and Applic ations , pp. 621–678. Cham: Springer International Publishing, 2014. [5] B. A. Moser, “Geometric characterization of Weyl’s discrepancy norm in terms of its n -dimensional unit balls,” Discr ete and Computational Ge ometry , vol . 48, no. 4, pp. 793–806, 2012. [6] B. A. Moser and T. Natschl¨ ager, “On stability of distance measures for ev ent sequences induced by level-crossing sampling.,” IEEE T r ansac- tions on Signal Pr o c essing , vol. 62, no. 8, pp. 1987–1999, 2014. [7] M. C. W. v an Rossum, “A nov el spike distance,” Neur al Computation , v ol. 13, no. 4, pp. 751–763, 2001. [8] J. D. Victor and K. P . Purpura, “Nature and precision of temp oral co ding in visual cortex: a metric-space analysis.,” Journal of Neur o- physiolo gy , vol. 76, pp. 1310–1326, Aug. 1996. [9] M. M. Deza and E. Deza, Encyclop e dia of Distanc es . Springer Berlin Heidelb erg, 2009. [10] H. Neunzert and B. W etton, “P attern recognition using measure space metrics,” T ech. Rep. 28, Univ ersity of Kaiserslautern, Department of Mathematics, Nov em b er 1987. [11] P . Bauer, U. Bodenhofer, and E. P . Klemen t, “A fuzzy system for image pixel classification and its genetic optimization,” in Cyb ernetics and Systems ’96 (R. T rappl, ed.), v ol. 1, (Vienna), pp. 285–290, Austrian So ciet y for Cyb ernetic Studies, April 1996. [12] B. A. Moser, “A similarity measure for image and v olumetric data based on Hermann Weyl’s discrepancy ,” IEEE T r ans. Pattern Analysis and Machine Intel ligenc e , vol. 33, no. 11, pp. 2321–2329, 2011. 17 [13] B. A. Moser, “The range of a simple random walk on Z : An elemen- tary combinatorial approac h,” The Ele ctr onic Journal of Combinatorics (EJC) , vol. 21, no. 4, p. P4.10, 2014. [14] W. F eller, “T he asymptotic distribution of the range of sums of inde- p enden t random v ariables,” Ann. Math. Statist. , vol. 22, pp. 427–432, 1951. [15] D. T. Qoyyimi and R. Zitikis, “Measuring the lac k of monotonicity in functions,” Mathematic al Scientist. , vol. 39, no. 2, pp. 107–117, 2014. [16] A. Alexiewicz, “Linear functionals on denjo y-integrable functions,” Col- lo quium Mathematic ae , v ol. 1, no. 4, pp. 289–293, 1948. Ac kno wledgmen t The author would lik e to thank the Austrian COMET Program and in particular Florian Sobieczky for careful reviewing and fruitful discussions. 18