### Table 1: Four convex loss functions and the corresponding -transform. On the interval [ B; B], each loss function has the indicated Lipschitz constant LB and modulus of con- vexity ( ) with respect to d . All have a quadratic modulus of convexity.

2004

"... In PAGE 3: ...) It is immediate from the definitions that ~ and are nonnegative and that they are also con- tinuous on [0; 1]. We calculate the -transform for exponential loss, logistic loss, quadratic loss and truncated quadratic loss, tabulating the results in Table1 . All of these loss func- tions can be verified to be classification-calibrated.... In PAGE 7: ...dometric d on a0 : we say that : a0 ! a0 is Lipschitz with respect to d, with constant L, if for all a; b 2 a0 , j (a) (b)j L d(a; b): (Note that if d is a metric and is convex, then necessarily satisfies a Lipschitz condition on any compact subset of a0 .) We consider four loss functions that satisfy these conditions: the exponential loss function used in AdaBoost, the deviance function for logistic regression, the quadratic loss function, and the truncated quadratic loss function; see Table1 . We use the pseudometric d (a; b) = inf fja j + j bj : constant on (minf ; g; maxf ; g)g : For all except the truncated quadratic loss function, this corresponds to the standard metric on a0 , d (a; b) = ja bj.... ..."

Cited by 9

### Table 1: Directory of signed distance functions and bounds.

1996

"... In PAGE 8: ...the distance to other shapes can be quite difficult. Table1 lists the primitives and operations for which the appendices contain signed distance functions and bounds. The Lipschitz constant is a useful quantity for deriving signed distance bounds to complex shapes.... ..."

Cited by 41

### Table 1. Directory of signed distance functions and bounds

"... In PAGE 3: ... 7, then f is a signed distance function. Table1 lists the primitives and operations for which the appendices contain signed distance functions and bounds. The Lipschitz constant is a useful quantity for deriving signed distance bounds to complex shapes.... ..."

### Table 1: Quantization versus Monte Carlo in 4-dimension.

2003

"... In PAGE 10: ... Then, in order to measure the error induced by the quantization in the scale of the MC estimator Standard Deviation, we wrote down the ratio absolute error (\ gi(Z)N ) . The results in Table1 illustrate a widely observed phenomenon when integrating func- tions by quantization: difierence of convex functions behave better than convex functions (this is obviously due to (2.17)), and Lipschitz derivative functions behave better than Lip- schitz continuous functions (as predicted by (2.... ..."

### Table I. Differentiable problems. Lipschitz constants and global solutions.

### Table II. Non-differentiable problems. Lipschitz constants and global solutions.

### Table 2. Regularization of functions with step discontinuities N(x) N(x; quot;)

"... In PAGE 7: ...7 Note that the last two formulas are independent of ; the particular choice indicated is numerically stable and allows to restrict the sum to those terms where the exponent is gt; log macheps, where macheps is the machine accuracy. To approximate elementary functions with step discontinuities, such as pos(x) = 1 if x gt; 0; 0 if x 0; nneg(x) = 1 if x 0; 0 if x lt; 0; sgn(x) = 8 lt; : 1 if x gt; 0; 0 if x = 0; ?1 if x lt; 0; one may use the regularizations given in Table2 (using x+ = max(x; 0)). How- ever, no accompanying theory is available since the results presented in this paper require the Lipschitz continuity of the functions involved.... ..."

### Table 2. Regularization of functions with step discontinuities N(x) N(x; quot;)

"... In PAGE 7: ...7 Note that the last two formulas are independent of ; the particular choice indicated is numerically stable and allows to restrict the sum to those terms where the exponent is gt; log macheps, where macheps is the machine accuracy. To approximate elementary functions with step discontinuities, such as pos(x) = 1 if x gt; 0; 0 if x 0; nneg(x) = 1 if x 0; 0 if x lt; 0; sgn(x) = 8 lt; : 1 if x gt; 0; 0 if x = 0; ?1 if x lt; 0; one may use the regularizations given in Table2 (using x+ = max(x; 0)). How- ever, no accompanying theory is available since the results presented in this paper require the Lipschitz continuity of the functions involved.... ..."

### Table 3: Example of common functions of devices: Same functions are mapped to the same gesture; similar functions may be mapped to the same gesture if this is intuitive and no other function is overloaded.

1997

"... In PAGE 13: ... Depending on the gesture commands are sent to the devices and feedback is applied. The correlation of gestures and commands is shown in Table3 . If the dialogue nishes by time out or by the pointing gesture or a certain termination gesture the control ow enters the direction determination (Figure 12).... In PAGE 30: ...evice at a time. First a device is selected by the unique pointer click. Depending on the selected device, the gesture will execute a certain command. Table3 shows a possible mapping. The stars indicate that the device in that column supports the function of the row.... ..."

Cited by 4

### Table 1: Example of common functions of devices: Same functions are mapped to the same gesture; similar functions may be mapped to the same gesture if this is intuitive and no other function is overloaded.

1997

"... In PAGE 2: ... 1). Further control is ac- cording to Table1 , where one gesture is used for each line. Every gesture is mapped to several similar tasks from di erent devices2, which reduces the number of gestures and makes the dialogue more intuitive.... ..."

Cited by 6