1 Introduction
Over the past decade, datasets for building and evaluating visual recognition systems have increased both in size and variation. The size of datasets has increased from a few hundred images to millions of images and the number of categories within the datasets has increased from tens of categories to more than a thousand categories. Coevolution of rich classification models along with advances in datasets have resulted in many commercial applications [10, 46, 33]. A multitude of operational challenges are posed while porting recognition systems from controlled lab environments to real world. A recognition system in the “open world” has to continuously update with additional object categories, be robust to unseen categories and have minimum downtime. Despite the obvious dynamic and open nature of the world, a vast majority of recognition systems assume a static and closed world model of the problem where all categories are known a priori. To address these operational issues, this paper formalizes and presents steps towards the problem of open world recognition. The key steps of the problem are summarized in Fig. 1.
As noted by [39], “when a recognition system is trained and is operational, there are finite set of known objects in scenes with myriad unknown objects, combinations and configurations – labeling something new, novel or unknown should always be a valid outcome”. One reason for the domination of “closed world” assumption of today’s vision systems is that matching, learning and classification tools have been formalized as selecting the most likely class from a closed set. Recent research, [39, 38, 16], has reformalized learning for recognition as open set recognition. However, this approach does not explicitly require that inputs be as known or unknown. In contrast, for open world recognition, we propose the system to explicitly label novel inputs as unknown and then incrementally incorporate them into the classifier. Furthermore, open set recognition as formulated by [39] is designed for traditional onevsall batch learning scenario. Thus, it is open set but not incremental and does not scale gracefully with the number of categories.
While there is a significant body of work on incremental learning algorithms that handle new instances of known classes [4, 5, 51]
, open world requires two more general and difficult steps: continuously detecting novel classes and when novel inputs are found updating the system to include these new classes in its multiclass open set recognition algoritim. Novelty detection and outier detection are complex issues in their own right with long histories
[29, 15] and are still active vision research topics [3, 28]. After detecting a novel class, the requirement to add new classes leaves the system designer with the choice of retraining the entire system. When the number of categories are small, such a solution may be feasible, but unfortunately, it does not scale. Recent studies on ImageNet dataset using SVMs or CNN require days to train their system [34, 19], e.g. 56 CPU/GPU days in case of CNN for 1000 category image classification task. Distance based classifiers like Nearest Class Mean (NCM) [17, 31, 36]offer a natural choice for building scalable system that can learn new classess incrementally. In NCMlike classifiers, incorporating new images or classes in implies adjusting the existing means or updating the set of class means. However, NCM classifier in its current formulation is not suited for open set recognition because it uses closeset assumptions for probability normalization. Handling unknowns in open world recognition requires gradual decrease in the value of probability (of class membership) as the test point moves away from known data into open space. The Softmax based probability assignment used in NCM does not account for open space.
The first contribution of this paper is a formal definition of the problem of open world recognition, which extends the existing definition of open set recognition which was defined for a static notion of set. In order solve open world recognition, the system needs be robust to unknown classes, but also be able to move through the stages and knowledge progression summarized in Fig. 1. Second contribution of the work is a recognition system that can continuously learn new object categories in an open world model. In particular, we show how to extend Nearest Class Mean type algorithms (NCM) [31], [36], to a Nearest NonOutlier (NNO) algorithm that can balance open space risk and accuracy.
To support this extension, our third contribution is showing that thresholding sums of monotonically decreasing functions of distances of linearly transformed feature space can have arbitrarily small “open space risk”. Finally, we present a protocol for evaluation for open world recognition, and use this protocol to show our NNO algorithm perform significantly better on open world recognition evaluation using ImageNet [2].
2 Related Work
Our work addresses an issue that is related to and has received attention from various communities such as incremental learning, scalable and open set learning.
Incremental Learning: As SVMs rose to prominence in many object recognition [52, 25], many incremental extensions to SVMs were proposed. Cauwenberghs et al. [4] proposed an incremental binary SVMs by means of saving and updating KKT conditions. Yeh et al. [51] extended the approach to object recognition and demonstrated multiclass incremental learning. Pronobis [35] proposed memorycontrolled online incremental SVM for visual place recognition. Although incremental SVMs might seem like a natural for large scale incremental learning for object recognition, they suffer from multiple drawbacks. The update process is extremely expensive (quadratic in the number of training examples learned [21]
) and depends heavily on the number of support vectors stored for performing updates
[21]. To overcome the update expense, [5] and [41] proposed classifiers with fast and inexpensive update process along with their multiclass extensions. However, the multiclass incremental learning methods and other incremental classifiers, [5, 41, 49, 24], are incremental in terms of additional training samples.Scalable Learning: Researchers like [30, 26, 11] have proposed label tree based classification methods to address scalability (# of object categories) in large scale visual recognition challenges [45, 2]
. Recent advances in deep learning community
[18], [43] has resulted in state of the art performance on these challenges. Such methods are extremely useful when the goal is obtain maximum classification/recognition performance. These systems assume a priori availability of entire training data (images and categories). However, adapting such methods to a dynamic learning scenario becomes extremely challenging. Adding object categories requires retraining the entire system, which could be infeasible for many applications. Thus, these methods are scalable but not incremental (Fig 2)Open Set Learning: Open set recognition assumes that there is incomplete knowledge of the world is present at training time, and that unknown classes can be submitted to an algorithm during testing [23, 39]. Scheirer et al. [39] formulated the problem of open set recognition for static onevsall learning scenario by balancing open space risk while minimizing empirical error. Scheirer et al. [38, 16] extended the work to multiclass settings by introducing compact abating probability model. Their work offers insights into building robust methods to handle unseen categories. However, class specific Weibull based calibration of SVM decision scores does not scale. Fragoso et al. [13] proposed a scalable Weibull based calibration for hypothesis generation for modeling matching scores, but do not address it in the context of general recognition problem.
The final aspect of related work is nearest class mean (NCM) classifiers. NCM classification, in which samples undergo a Manhanalobis transform and then are are associated with a class/cluster mean, is a classic pattern recognition approach
[14]. NCM classifiers have a long history of use in vision systems, [7] and have multiple extensions, adapations and applications [9, 44, 50, 20, 27]. Recently the technique has been adapted for use in larger scale vision problems [48, 47, 31, 36], with the most recent and most accurate approaches combining NCM with metric learning [31]and with random forests
[36].Since we extend NCM classification, we briefly review the formulation including a probabilistic interpretation. Consider an image represented by a dimensional feature vector . Consider object categories with their corresponding centroids , where . Let be images for each object category. The centroid is given by . NCM classification of a given image instance with a feature vector is formulated as searching for the closest centroid in feature space as . Here represents a distance operator usually in Euclidean space. Mensink et al. [31] replace Euclidean distance with a lowrank Mahalanobis distance optimized on training data. The Mahalanobis distance is induced by a weight matrix , where D is the dimensionality of the lower dimensional space. Class conditional probabilities
using an NCM classifier are obtained using a probabilistic model based on multiclass logistic regression as follows:
(1) 
In the above formulation, class probabilities are set to be uniform over all classes. During Metric learning optimization, Mensink et al. [31] considered nonuniform probabilities given by:
(2) 
where Z denotes the normalizer and is a per class bias.
3 Open World Recognition
We first establish preliminaries related to open world recognition, following which we formally define the problem. Let classes be labeled by positive integers and let be the set of labels of known classes at time . Let the zero label () be reserved for (temporarily) labeling data as unknown. Thus includes unknown and known labels.
Let our features be . Let be a measurable recognition function, i.e. implies recognition of the class of interest and when is not recognized, where is a suitably smooth space of recognition functions.
The objective function of open set recognition, including multiclass formulations, must balance open space risk against empirical error. As a preliminary we adapt the definition of open space and open space risk used in [39]. Let open space, the space sufficiently far from any known positive training sample , be defined as:
(3) 
where is a closed ball of radius centered around any training sample . Let be a ball of radius that includes all known positive training examples as well as the open space . Then probabilistic Open Space Risk for a class can be defined as
(4) 
That is, the open space risk is considered to be the relative measure of positively labeled open space compared to the overall measure of positively labeled space.
Given an empirical risk function , e.g. hinge loss, the objective of open set recognition is to find a measurable recognition function that manages (minimizes) the Open Set Risk:
(5) 
where is a regularization constant.
With the background in place, we formalize the problem of open world recognition.
Definition 1 (Open World Recognition):
A solution to open world recognition is a tuple with:

A multiclass open set recognition function using a vector function of perclass measurable recognition functions , also using a novelty detector . We require the per class recognition functions for to be open set recognition functions that manage open space risk as Eq.4. The novelty detector determines if results from vector of recognition functions is from an unknown () class.

A labeling process applied to novel unknown data from time , yielding labeled data where . Assume the labeling finds new classes, then the set of known classes becomes .

An incremental learning function to scaleably learn and add new measurable functions , each of which manages open space risk, to the vector of measurable recognition functions.
Ideally all of these steps should be automated, but herein we presume supervised learning with labels obtained by human labelling.
If we presume that each reports a likelihood of being in class and that we presume normalized across the respective classes and Let . For this paper we let the multiclass open set recognition function be given as
(6)  
(7) 
With these definitions a simple approach for the novelty detection is to set a minimum threshold for acceptance, e.g. letting . In the following section we will prove this simple approach can manage open space risk and hence provide for item 1 in the open world recognition definition.
4 Opening existing algorithms
The series of papers [39, 38, 16] formalized the open set recognition problem and proposed 3 different algorithms for managing open set risk. It is natural to consider these algorithms for open world recognition. Unfortunately, these algorithms use EVTbased calibration of 1vsrest RBF SVMs and hence are not well suited for incremental updates or scalability required for open world recognition. In this paper we pursue an alternative approach better suited to open world using nonnegative combinations of abating distance. Using this approach Sec 4.1 shows that NCM can be inexpensively extended to open world recognition, which is termed as Nearest NonOutlier (NNO) algorithm.
The authors of [38] show that if a recognition function is decreasing away from the training data, a property they call abating, then thresholding the abating function limits the labeled region and hence can manage/limit open space risk. The Compact Abating Probability (CAP) model presented in that paper is a sufficient model, but it is not necessary. In particular we build on the concept of a CAP model but generalize the model showing that any nonnegative combination of abating functions, e.g., a convex combination of decreasing functions of distance, can be thresholded to have zero open space risk. We further show that we can work in linearly transformed spaces, including projection onto subspaces, and still manage open space risk and that NCM type algorithms manage open space risk.
Theorem 1 (Open space risk for model combinations):
Let be a recognition function that thresholds a nonnegative weighted sum of CAP models ( ) over a known training set for class , where and is a CAP model. Then for s.t. , i.e. one can threshold the probabilities to limit open space risk to any desired level.
Proof: It is sufficient to show the condition holds for , since similar to Corollary 1 of [38], larger values of may simply allow larger labeled regions with larger open space risk. Considering each model separately, we can apply Theorem 1 of [38] to each yielding a such that the function defines a labeled region with zero open space risk. Letting it follows that is contained within , which as a finite union of compact regions with zero risk, is itself a compact labeled region with zero open space risk. Q.E.D
The theorem/proof trivially holds for a max over classes. The proof can be generalized to combinations via product. The proof can also be generalized to combinations of monotonic transformed recognition functions, with appropriate choice of thresholds, but for this paper we need only a max or sum of models. However, we also need to work in transformed spaces especially in lowerdimensional projected spaces.
Theorem 2 (Open Space Risk for Transformed Spaces):
Given a linear transform let , yields a linearly transformed space of features derived from feature space . Let be the transformation of points in open space . Let be a probabilistic CAP recognition function over and let be a recognition function over . Then , i.e. managing open set risk in will also manage it in the original feature space .
Proof: If is dimensionalty preserving, then the theorem follows from the linearity of integrals in the definition of risk. Thus we presume is projecting away dimensions. Since the open space risk in the projected space is we have where is the Lebesgue measure in and . Since , i.e. is contained within a ball of radius , it follows from the properties of Lebesgue measure that and hence the open space risk in is bounded. Q.E.D.
It is desirable for open world problems that we consider the error in the original space. We note that varies with dimension and the above bounds are generally not tight. While the theorem gives a clean bound for zero open space risk, for a solution with nonzero risk in the lower dimensional space, when considered in the original space, the solution may have open space risk that increases exponentially with the number of missing dimensions.
We note that these theorems are not a license to claim that algorithms, with rejection, manage open space risk. While many algorithms can be adapted to compute a probability estimate of per class inclusion and can threshold those probabilities to reject, not all such algorithms/rejections manage open space risk. Thresholding Eq
2, which [31] minimizes in place of 1, will not work because the function does not always decay away from known data. Similarly, rejecting decision close to the plane in a linear SVM does not manage open space risk, nor does the thresholding layers in a convolution neural network
[40].On the positive side, these theorems show that one can adapt algorithms that linearly transforms feature space and use a probability/score mapping that combines positive scores that decrease with distance from a finite set of known samples. In the following section, we demonstrate how to generalize an existing algorithm while managing open space risk. Open world performance, however, greatly depends on the underlying algorithm and the rejection threshold. While theorems 1 and 2 say there exists a threshold with zero open space risk, at that threshold there may be minimal or no generalization ability.
4.1 Nearest NonOutlier (NNO)
As discussed previously (sec 1), one of the significant contributions of this paper is combining theorems 1 and 2 to provide an example of open space risk management and move toward a solution to open world recognition. Before moving on to defining open world NCM, we want to add a word of caution about “probability normalization” that presumes all classes are known. e.g. softmax type normalization used in eqn 1. Such normalization is problematic for open world recognition, where there are unknown classes, In particular,
in open world recognition the Law of Total Probability and Bayes’ Law cannot be directly applied
and hence cannot be used to normalize scores. Furthermore, as one adds new classes, the normalization factors and hence probabilities, keep changing and thereby limiting interpretation of the probability. For an NCM type algorithm, normalization with the softmax makes thresholding very difficult since for points far from the class means the nearest mean will have a probability near 1. Since it does not decay, it does not follow Theorem 1.To adapt NCM for open world recognition we introduce Nearest NonOutlier (NNO) which uses a measurable recognition function consistent with Theorems 1 and 2. Let NNO represent its internal model as a vector of means . Let be the linear transformation dimensional reduction weight matrix learned by the process described in [31]. Then given , let
(8) 
be our measurable recognition function with giving the probability of being in class in class , where standard gamma function which occurs in the volume of a mdimensional ball. Let with and given by Eq. 7. Let with .
That is, NNO rejects as an outlier for class when , and labels input as unknown/novel when all classes reject the input. Finally, after collecting novel inputs, let the human labeled data for a new class and let our incremental class learning compute and append to .
Corollary 1 (NNO solves open world recognition):
The NNO algorithm with human labeling of unknown inputs is a tuple , consistent with Definition 1, hence NNO is a open world recognition algorithm.
5 Experiments
In this section we present our protocol for open world experimental evaluation of NNO, and a comparison to NCM based classifiers.
Dataset and Features: Our evaluation is based on the ImageNet Large Scale Visual Recognition Competition 2010 dataset. ImageNet 2010 dataset is a large scale dataset with images from 1K visual categories. The dataset contains 1.2M images for training (with around 660 to 3047 images per class), 50K images for validation and 150K images for testing. Large number of visual categories allow us to effectively gauge performance of incremental and open world learning scenarios. In order to effectively conduct experiments using open set protocol, we need access to ground truth. ILSVRC’10 is the only ImageNet dataset will full ground truth, which is why we selected that dataset over later releases of ILSVRC (e.g. 20112014).
We used densely sampled SIFT features clustered into 1K visual words as given by Berg et al. [2]. Though more advanced features are available [34, 19, 42], extensive evaluation across features is beyond the scope of this work ^{1}^{1}1In the supplemental material we present some experiments on additional features on ILSVRC’13 data so show the advantages of NNO are not feature dependent
. Each feature is whitened by its mean and standard deviation to avoid numerical instabilities. We report performance in terms of average classification accuracy obtained using top1 accuracy as per the protocol provided for the ILSVRC’10 challenge. As our work involves initially training a system with small set of visual categories and incrementally adding additional ategories, we shun top5 accuracy.
Algorithms: We use code provided by Mensink et al. [31] as the baseline. This algorithm has near state of the art results and while recent extension with random forests[36] improved accuracy slightly, [36] does not provide baseline code. Since we are primarily focused on open world aspects, the NCM baseline using the original authors code provide a sufficient baseline. The baseline NCM algorithm is evaluated using closed set (CSNCM) and open set (OSNCM) in incremental learning phase. We also report performance on our Nearest NonOUTLIER (NNO) extension of NCM classifier in both closeset testing (CSNNO) and Open set testing (OSNNO).
5.1 Open World Evaluation Protocol
Closed set evaluation is when a system is tested with all objects known during testing, i.e. training and testing use the same classes but different instances. In open set evaluation, the system is tested with examples from both known and unknown categories, where unknown categories are categories not used during training. Open set recognition evaluation protocol proposed by by Scheirer et al. [39] does not handle the open world scenario in which object categories are being added to the system continuously. Ristin et al. [36] presented an incremental closed set learning scenario where novel object categories are added continuously. We combined ideas from both of these approaches and propose a protocol that is suited for open world recognition in which categories are being added to the system continuously while the system is also tested with unknown categories.
Training Phase: The training of the NCM classifier is divided into two phases: an initial metric learning/training phase and a growth/incremental learning phase. In the metric learning phase, a fixed set of object categories are provided to the system. The system performs parameter optimization including metric learning on these categories. Once the metric learning phase is completed, the incremental learning phase uses the fixed metrics and parameters. During the incremental learning phase, object categories are added to the system onebyone. While for scaleability one might measure time, both NCM and NNO add new categories in the same way and it is extremely fast, since it only consists of computing the means, so we don’t report/measure timing here.
Nearest Non Outlier (NNO) is our extension of NCM classifier based on the CAP model requires estimation of for eq. 8. This is done in the parameter estimation phase using the metric also learned in that phase. The validation data for training phase is divided in two sets: known categories and unknown categories. A for NNO is estimated over the training known and unknown categories by optimizing for F1measure. This process is repeated over multiple folds and the average is obtained. During evaluation process, the average is used and thresholding at zero determines if the incoming image belongs to an unknown category.
Testing Phase: To ensure proper open world evaluation, we split the ImageNet test data into two sets of 500 categories each: the known set and the unknown set. At every stage, the system is evaluated with a subset of the known set and the unknown set to obtain closed set and open set performance. This process is repeated as we continue to add categories to the system. The whole process is repeat ed across multiple dataset splits to ensure fair comparisons and estimate error. While [39] suggest a particular openness measure, it does not address the incremental learning paradigm. We fixed the number of unknown categories and report performance as series of known categories are incrementally added. We present separate plots for different number of unknown categories.
Multiclass classification error [6] for a system trained with test samples is given as For open world testing the evaluation must keep track of the errors which occur due to standard multiclass classification over known categories as well as errors between known and unknown categories. Consider evaluation of samples from known categories and samples from unknown categories leading to test samples and . Thus, open world error for a system trained over categories is given as:
(9) 
5.2 Experimental Results
We now compare performance of CSNCM, OSNCM and OSNNO algorithms. In first experiment, we perform metric learning on a relatively few (50) categories to study the validity of the proposed approach. We obtained closed set classification results using CSNCM which serves as our baseline. We add 50 categories incrementally, by updating the system with means of the incoming categories. To obtain closed set performance, we perform testing with 50, 100, 150 and 200 categories respectively. Table 1 shows the number of training and testing categories used in Fig 2(a). These categories are same as the ones used for training. Open set performance is obtained by considering an additional 100 unknown categories for testing leading to overall testing with 150, 200, 250 and 300 categories respectively. The results of this experiment are shown in Fig 3. As we add categories to the system in close set testing, the performance of both CSNCM and CSNNO drop rather gracefully, which is expected. However, in case of open set testing OSNCM, the performance drop is drastic because of the unknown categories which NCM was not designed to handel. More formally, the second term in Eqn 9 dominates the error encountered by the system. There is perforamce drop from CSNNO to OSNNO, but nowhere near as dramatic as the NCM drop. When the same set of test data is testing with our OSNNO model, we see significant performance improvement gains over OSNCM. We repeat similar experiment with examples from 200 and 500 unknown categories. We observe OSNNO consistently performs well on open world recognition task.
Metric Learning  Incremental Learning  


50  100  150  200  

50  100  150  200  





In second experiment, we consider 200 categories for metric learning and parameter estimtion, and successively add 100 categories in the incremental learning phase. By the end of the learning process, the system needs to learn a total of 500 categories. Open set evaluation of the system is carried out with 100, 200 and 500 unknown categories with results show in Figs 3(a), 3(b) and 3(c) respectively. In final stage of the learning process i.e 500 categories for training and 500 (known) + 500 (unknown) categories for open set testing (Fig 3(c) ), we use all the categories from ImageNet (1000) for our evaluation process. We observe similar rankordering of algorithms in this experiment as in the previous experiment. On the largest scale task involving 500 categories in training and 1000 categories in testing, we observe almost 74% improvement of OSNNO over OSNCM. We repeated the above experiments over multiple folds and found the standard deviation across folds to be on the order of 1% which is not visible in the figure.
The training time required for the initial metric learning process depends the SGD speed and convergence rate. We used close to 1M iterations which resulted in metriclearning time of 15 hours in case of 50 categories and 22 hours in case of metric learning for 200 categories. Given the metric, the learning of new classes via the update process is extremely fast as it is simply computation of means from labeled data. The majority of time in update process is dominated by feature extraction and then file I/O, but could easily be in real time. Multiclass recognition, including detecting novel classes, is also easily done in real time.
6 Discussion
In this work, we formalized the problem of open world recognition, and provide an open world evaluation protocol. We extended existing work on NCM classifiers and showed formally how they can be adapted for open world recognition. The proposed NNO algorithm consistently outperforms NCM on open world recognition tasks and is comparable to NCM on closed set – we gain robustness to the open world without much sacrifice.
There are multiple implications of our experiments. First, we demonstrates suitability of NNO for large scale recognition tasks in dynamic environments. NNO allows construction of scalable systems that can be incrementally add classes and that are robust to unseen categories. Such systems are suitable where minimum down time is desired.
Second, as can be seen in Figs. 3, 4 as the number of categories known to the system increases, OSNNO remains relatively stable but the closed set performance for CSNCM and CSNNO quickly reduces down to the performance open set. This suggests incrementally adding more classes in the system is limited by open space risk and that closed set recognition problem becomes similar to open world recognition problem. We conjecture that as the number of classes grow the close world converges to an open world and thus open world recognition is a natural setting for building scalable systems.
While we provide one viable approach to extension, the theory herein allows a broad range of approaches; improved CAP models and better open set probability calibration should be explored.
Open world evaluation across multiple features for a variety of applications is an important future work. Recent advances in deep learning and other areas of visual recognition have demonstrated significant improvements in absolute performance. The best performing systems on such tasks use parallel system and train for days. Extending these to incremental open world performance, one may be able to reuse the deeply learned features with a top layer of open world multiclass to provide a hybrid solution. While scalable learning in open world is critical for deploying computer vision applications in the real world, high performing systems enable adoption by masses. Pushing absolute performance on large scale visual recognition challenges
[2] development of scalable systems for open world are essentially two sides of the same coin.7 Supplemental Material : Towards Open World Recognition
In this supplemental section, we provide additional material to further the reader’s understanding of the work on open world recognition, CAP models and the Nearest NonOutlier algorithm that we present in the main paper. We present additional experiments on ILSVRC 2010 dataset. We then present experiments on ILSVRC 2012 dataset to demonstrate that performance gain of OSNNO over CSNCM (see fig 3 and 4 in the main paper) are not feature/dataset specific. Finally first provide algorithmic pseudocode for implementing the NNO algorithm.
7.1 Experiments on ILSVRC 2010
7.1.1 Thresholding NCMSoftmax for ILSVRC 2010
In section 4.1 of the main paper, we explain the process of rejecting samples from unseen categories to balance open space risk and defined in Eq. 8, a probability function which is thresholded at zero. At first it might seem like a viable idea to just threshold the original softmax probability used in NCM. As explained in the main paper this will fail for open set because the normalization is improper and hence the softmax probability calibration will bias results. To convince the skeptical reader, we add a small experiment, similar to fig 3a in the main paper, and show the performance of classifying samples as unknown by directly thresholding softmax probabilities. The reader can observe the performance of OSNCMSTH is similar to OSNCM and significantly worse than OSNNO. Just thresholding the softmax probability is not enough, because its normalization keeps it from decaying as one move away from known data. This result confirms the suitability of balancing open set risk with Eq 8,using transformed learned Mahalanobis distance to the NCM. The results from this experiment are shown in fig 5. Table 1 lists the different algorithms used.
7.1.2 Performance of NNO for different values of
Section 4.1 and 5.1 in the main paper describes NNO algorithm in detail and steps involved in estimating optimal required to balance open space risk. Section Alg 1 illustrates steps involved in developing NNO algorithm for open set. In the experimental results shown in Fig 3 and 4 in the main paper, we used optimal for evaluation purpose, which was approximately 5000. In this section, we show the effect of different values of on the performance of OSNCM to give the reader a feeling for the sensitivity of that parameter. The optimal value is part of a broad peak, and small changes in have minimal impact. Even changing it by 20% has only a small impact on open set testing. These results are illustrated with respect to fig 3a in the main paper. In our experiments, we observed similar trends for all other experiments.
Fig 6 shows performance for varying set of . is the optimal threshold that was selected. We observe that performance of OSNNO continues to improve as we near the optimal threshold. For a threshold value lower than , the number of correct predictions retained reduces significantly. Thus, a balance between correct predictions retained and unknown categories rejected has to be maintained. This balance is maintained by the selected .
Notation  Algorithm  

CSNCM 


OSNCM 


CSNNO 


OSNNO 


OSNCMSTH 

7.2 Experiments on ILSVRC 2012 Dataset
As noted in the section 5 (Experiments) in the main paper, we used ILSVRC 2010 dataset because we needed access to ground truth to for the test set. Ground truth is was necessary to perform the open world recognition test protocol, which includes selecting known and unknown set of categories. In this section, we perform additional experiments on ILSVRC 2012 [37] ^{2}^{2}2ILSVRC dataset remained unchanged between 2012, 2013 and 2014 dataset across multiple features to show the effectiveness of NNO algorithm for closed set and open set tasks does not significantly depend on feature type.
Since ground truth is not available for ILSVRC’12 dataset, we split the training data provided by the authors into training and test split. The number of categories is the same, this just limits the number of images per class used. We use 70% of training data to train models and 30% of the data for evaluation. This process is repeated over multiple folds. Once the data is split into training and test split, remaining procedure for metric learning and incremental learning is followed similar to that in section 5 (Experiments) in the main section. We conduct similar 2 sets of experiments on ILSVRC’12 data: metric learning with 50 and 200 initial categories as shown in Figs 3 and 4 in the main paper. The closed set and open open set testing is conducted in similar manner as well. While the open world experimental setup for ILSVRC’12 is not ideal because of the smaller number of images per class, the goal of this experiment is to show that the advantages of NNO are not feature dependent.
We use precomputed features as provided on cloudcv.org [1]. We consider three set of features as follows:

Dense SIFT: SIFT descriptors are densely extracted [22]
using a flat window at two scales (4 and 8 pixel radii) on a regular grid at steps of 5 pixels. The three descriptors are stacked together for each HSV color channels, and quantized into 300 visual words by kmeans. The features used in the main paper are similar to these features, except that in the main paper, dense SIFT features were quantized into 1000 visual words by kmeans.

Histogram of Oriented Gradients (HOG): HOG features are used in wide range of visual recognition tasks [8]. HOG features are densely extracted on a regular grid at steps of 8 pixels. HOG features are computed using code provided by [12]. This gives a 31dimension descriptor for each node of the grid. Finally, the features are quantized into 300 visual words by kmeans.

Local Binary Patterns (LBP): Local Binary Patterns (LBP) [32]
is a texture feature based on occurrence histogram of local binary patterns. It has been widely used for face recognition and object recognition. The feature dimensionality used was 59.
Results using Dense SIFT, HOG and LBP features are shown in figures 6(a), 6(b) and 6(c) respectively. The absolute performance with Dense SIFT features is the best, followed by HOG and LBP. The Dense SIFT is very similarly to the results on ILSVRC 2010. Moreover, from these experiments we observe similar trends across all features to the trends seen in Figs 3 and 4 in the main paper. We see that as closed set performance of CSNCM and CSNNO is comparable while OSNCM suffers significantly when tested with unknown set of categories. We continue to see significant gains of OSNNO over OSNCM across HOG and dense SIFT features. We also observe the trend where as we add more categories in the system, the closed set and open set performance begin to converge. Thus, it is reasonable to conclude that the performance gain seen in terms of OSNNO is not feature dependent. These observations are consistent with our observations from experiments on ILSVRC’10 data.
7.3 Algorithmic Pseudocode for Nearest NonOutlier (NNO)
In this section, we provide pseudocode for Nearest NonOutlier algorithm as described in section 4.1 in the main paper. The algorithm proceeds in multiple steps. In the first step, features are normalized by the mean and standard deviation over the starting subset. The initial set of features is used to perform metric learning. Following this step, threshold for open set NNO is estimated using per class decisions using per Eq. 8 in the main paper and a cross class validation procedure of [16] training data splits. The complete pseudocode is given in Alg 1
7.4 Acknowledgement
We would like to thank Thomas Mensink (ISLA, Informatics Institute, University of Amsterdam) for sharing code and Marko Ristin (ETH Zurich) for sharing features. The work was carried out with the help of NSF Research Grant IIS1320956 (Open Vision  Tools for Open Set Computer Vision and Learning) and UCCS Graduate School Fellowship.
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