Jean-Eric Pelet

Basma Taieb

Rami Alkhudary

Measuring consumer perceptions of home-delivery convenience -the case of cargo bikes

Keywords: e-commerce, last-mile logistics, home delivery, cargo bike, service convenience, consumer perceptions, home-delivery convenience, HDCCB

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The growth of e-commerce and the intensifying demand for speed drive the need for innovation in the design, management, and operation of urban logistics systems. The resultant innovations should also be sustainable and cost effective [START_REF] Faugère | Dynamic pooled capacity deployment for urban parcel logistics[END_REF][START_REF] Risberg | Last mile practices in e-commerce : Framework development and empirical analysis of Swedish firms[END_REF]. As the increasing flow of e-commerce orders continues to generate new records for annual sales, the dominant players in the last-mile market face the challenges of increasing customer demand. In response, e-retailers and logistics service providers seek innovative solutions, which are often based on technological advances (Teller et al., 2019;Vakulenko et al., 2019).

By 2050, 86.6% of the population of the more developed regions of the world will be living in urban areas (UN DESA, 2018). In 2022, the degree of urbanization worldwide was at 57% (Population Reference Bureau, 2022). Increasing population density is a challenge for urban logistics because it results in traffic congestion and necessitates the use of irregular locations for logistics facilities. These problems become even more complex in megacities due to their high population density [START_REF] Fransoo | Reaching 50 million nanostores : Retail in distribution in emerging megacities[END_REF]. Online retail, which requires goods to be transported to consumers' homes, increases the incidence of freight movement within cities.

At the same time, the size of individual shipments decreases [START_REF] Savelsbergh | 50th Anniversary Invited Article-City Logistics : Challenges and Opportunities[END_REF], making the planning of last-mile logistics more difficult. In addition, online shippers must continue to meet consumers' expectations about speed (Manerba et al., 2018). In recent decades, last-mile delivery has attracted growing attention because of the e-commerce boom [START_REF] Ouyang | Community logistics and dynamic community partitioning : A new approach for solving e-commerce last mile delivery[END_REF]. However, to date, research on delivery by cargo bike (CB) remains extremely scarce. The uniqueness and the practical relevance of the present study therefore lie in its potential to reveal solutions to the problems of concentration and traffic. These solutions can enable the delivery of e-commerce products to remain convenient.

Last-mile logistics is a complex system that involves strategic planning. Such planning is necessary in order to implement and manage the efficient transportation and storage of goods from the receipt of an order to the end customer (Melkonyan et al., 2020;[START_REF] Olsson | Exploring past achievements in last mile logistics research : A literature review[END_REF].

Last-mile distribution activities focus on channel design, inventories, delivery, and execution at the tactical and the operational level. The last mile is defined as the final segment of the delivery process. That segment is related to a set of activities that are necessary for successful delivery from the last transit point to the last transfer point in the supply chain [START_REF] Lagin | Last-mile logistics of perishable products : A review of effectiveness and efficiency measures used in empirical research[END_REF][START_REF] Yuen | An investigation of customers' intention to use self-collection services for last-mile delivery[END_REF]. Last-mile deliveries are an increasingly important and costly component of supply chains, especially in urban centers (van Lopik et al., 2020). The last meters of many deliveries are covered on foot. Although new technologies that can support improved navigation are being explored, the focus, to date, has been on technical implementation (van Lopik et al., 2020).

As far as the sustainable development of cities is concerned, CBs can provide an attractive solution to some of the challenges of last-mile logistics. CBs, as a technology, lie at the intersection of micromobility and electric mobility. They are two-to-four-wheel vehicles that can carry cargo. An electric motor often assists the driver when they pedal [START_REF] Dalla Chiara | How cargo cycle drivers use the urban transport infrastructure[END_REF]. Many alternatives to CBs can be used in last-mile logistics, including drones [START_REF] Rejeb | Drones for supply chain management and logistics : A review and research agenda[END_REF] and electric autonomous vehicles (Monios & Bergqvist, 2020). CBs are more likely to be used for the last mile of a delivery in urban centers with narrow roads [START_REF] Fontaine | The vehicle routing problem with load-dependent travel times for cargo bicycles[END_REF][START_REF] Muñoz-Villamizar | Short-and mid-term evaluation of the use of electric vehicles in urban freight transport collaborative networks : A case study[END_REF]. The CB seems to be a viable solution for urban logistics and the transportation of light goods in inner cities. In environmental terms, CBs reduce road traffic and thus emissions of CO2. They emit no greenhouse gases and make little noise. Thus, using CBs actively reduces air and noise pollution [START_REF] Buldeo Rai | The "next day, free delivery" myth unravelled : Possibilities for sustainable last mile transport in an omnichannel environment[END_REF][START_REF] Muñoz-Villamizar | Short-and mid-term evaluation of the use of electric vehicles in urban freight transport collaborative networks : A case study[END_REF]. This perception may lead some consumers to prefer CBs over other delivery models. Many authors have confirmed the potential of CBs in urban environments (Melo & Baptista, 2017;[START_REF] Mühlbauer | A parallelised large neighbourhood search heuristic for the asymmetric two-echelon vehicle routing problem with swap containers for cargobicycles[END_REF]Sheth et al., 2019). Indeed, the companies that are involved in urban logistics are increasingly using CBs or evaluating them in pilot projects [START_REF] Leonardi | Before-After Assessment of a Logistics Trial with Clean Urban Freight Vehicles : A Case Study in London[END_REF].

Convenience is important for goods, services, ideas, and anything else that may add to one's comfort. It is a multidimensional construct that not only reveals strategic and tactical marketing opportunities [START_REF] Brown | The Strategic and Tactical Implications of Convenience in Consumer Product Marketing[END_REF] but can also have numerous implications for delivery companies in the context of last-mile logistics [START_REF] Colwell | Toward a measure of service convenience : Multiple-item scale development and empirical test[END_REF][START_REF] Reimers | A consumer definition of store convenience (finally)[END_REF][START_REF] Reimers | Retail centres : It's time to make them convenient[END_REF]. Convenience can be viewed as a variable that aggregates the four utilities of economic theory-a service is convenient if it is available in the right place and at the right time as well as if its acquisition, ownership, and use require little effort [START_REF] Brown | Convenience in Services Marketing[END_REF]. Despite the remarkable growth of e-commerce and the use of CBs in last-mile logistics, no study has explicitly examined the convenience of last-mile home delivery (HD) with CBs (Ma et al., 2022;[START_REF] Reimers | Spatial convenience : Bridging the gap between shopping malls and shopping strips[END_REF][START_REF] Salhieh | Integrating vehicle tracking and routing systems in retail distribution management[END_REF]. Therefore, our study is intended to 1) identify the dimensions of HD convenience (time, use, execution, and acquisition) and 2) to develop a reliable and valid measure of last-mile HD convenience in the context of CBs.

Accordingly, we present the Home-Delivery Convenience via Cargo Bike (HDCCB) scale. We introduce the four main dimensions on which convenience depends, namely use, time, execution, and acquisition, and we propose a weighting for them.

The paper is organized as follows: Section 2 describes the theoretical foundations of the study and explains the importance of considering the convenience of HD as a multidimensional construct in last-mile logistics with CB. Section 3 presents the methodology of the empirical study, and Section 4 presents the results. The conclusions, limitations, and avenues for future research are described in Section 5.

Literature review

In this section, we review the relevant literature to understand the potential of CBs in last-mile logistics and to identify the differences between the consumer perceptions of convenience services that have emerged from studies on marketing and e-retail.

Home delivery and the cargo bike in last-mile deliveries

The last mile is critical to the delivery of goods and services to consumers, accounting for 13% and 75% of total supply-chain costs (M. [START_REF] Wang | Augmented reality (AR) app use in the beauty product industry and consumer purchase intention[END_REF]. The technology and convenience of e-commerce are increasingly exerting pressure on the last-mile distribution system and creating a need for new logistics solutions. The discrete temporal and spatial distribution of customer demand leads to higher rates of error in deliveries and to lower vehicle utilization; consequently, last-mile deliveries have become time consuming and uncertain. CBs are a promising solution that is already used widely in some cities. A CB is a type of commercial electric vehicle. Critically, CBs are quicker than cars over short distances [START_REF] Fontaine | The vehicle routing problem with load-dependent travel times for cargo bicycles[END_REF][START_REF] Raeesi | Coordinated routing of electric commercial vehicles with intra-route recharging and en-route battery swapping[END_REF].

CBs can avoid traffic congestion, they can take shortcuts through parks, and they can be parked on sidewalks, minimizing walking distances [START_REF] Jie | The two-echelon capacitated electric vehicle routing problem with battery swapping stations : Formulation and efficient methodology[END_REF][START_REF] Rudolph | Cargo cycles in commercial transport : Potentials, constraints, and recommendations[END_REF]. Recent developments in bicycle technology have contributed to the use of bicycles for urban freight transportation. These developments include the use of lighter materials; the design of bicycles with larger cargo capacities, in terms of both weight and volume; the development of trailers; and the introduction of electric motors that assist the rider, especially on inclines [START_REF] Browne | Evaluating the use of an urban consolidation centre and electric vehicles in central London[END_REF]Mashalah et al., 2022). As a result, this type of transportation is finally being adopted in large metropolitan areas [START_REF] Goeke | Routing a mixed fleet of electric and conventional vehicles[END_REF][START_REF] Muñoz-Villamizar | Short-and mid-term evaluation of the use of electric vehicles in urban freight transport collaborative networks : A case study[END_REF].

One of the strongest ideas that pertain to the sustainable supply chain is the use of environmentally friendly vehicles for last-mile delivery [START_REF] Fontaine | Scheduled service network design with resource management for two-tier multimodal city logistics[END_REF]. Since a CB does not produce CO2, CBs are considered to offer a suitable alternative to traditional delivery methods [START_REF] Buldeo Rai | The "next day, free delivery" myth unravelled : Possibilities for sustainable last mile transport in an omnichannel environment[END_REF]. Unlike conventional bicycles, CBs have trunks. These trunks can be as large as 2 m 3 , and they can hold up to 250 kg of goods. In addition, CBs are equipped with electric motors. Their use seems to be particularly important because modern consumers are concerned about the environment, which makes them more favorably disposed to the service. Consequently, its provision presents delivery companies with both financial and reputational opportunities (Louis et al., 2019).

Research on electric vehicles in the supply chain often lacks a theoretical background.

Many studies rely heavily on game theory [START_REF] Gu | Subsidising an electric vehicle supply chain with imperfect information[END_REF][START_REF] Wu | Optimal postponement contracting decisions in crowdsourced manufacturing : A three-level game-theoretic model for product family architecting considering subcontracting[END_REF][START_REF] Zhang | Recycling mode selection and carbon emission reduction decisions for a multi-channel closed-loop supply chain of electric vehicle power battery under cap-and-trade policy[END_REF], with a minority of contributions drawing on resource dependence theory [START_REF] Kalaitzi | Managing resource dependencies in electric vehicle supply chains : A multi-tier case study[END_REF], credibility theory (S. [START_REF] Zhang | Fuzzy optimization model for electric vehicle routing problem with time windows and recharging stations[END_REF]), graph theory (Miao et al., 2014), stakeholder theory [START_REF] Bhuyan | Evaluating the lithium-ion battery recycling industry in an emerging economy : A multi-stakeholder and multicriteria decision-making approach[END_REF], innovation diffusion theory [START_REF] Ruoso | An assessment of barriers and solutions for the deployment of electric vehicles in the Brazilian market[END_REF], grounded theory [START_REF] Anosike | Exploring the challenges of electric vehicle adoption in final mile parcel delivery[END_REF], practice theory (Sopjani et al., 2020), and transition theory (Sushandoyo & Magnusson, 2014). In order to integrate consumers into the supply chain more seamlessly [START_REF] Ageron | Connecting the customer to the supply chain : Proposition of a conceptual framework[END_REF], we adopt a novel theoretical approach-our objective is to identify dimensions of the convenience of services [START_REF] Brown | The Strategic and Tactical Implications of Convenience in Consumer Product Marketing[END_REF][START_REF] Brown | Convenience in Services Marketing[END_REF] and to create a measure of these dimensions for last-mile logistics with CBs. [START_REF] Copeland | Relation of Consumers : Buying Habits to Marketing Methods[END_REF] suggested that the time and the physical and mental effort that the purchase of a product require from the consumer should be minimized. [START_REF] Brown | Convenience in Services Marketing[END_REF] defined service convenience as any facilitation that causes the consumer to experience excitement about a particular service or product. For example, the service or product may be sold in a way that conserves the time and the effort that the consumer must expend. In order to provide a highquality service that meets consumer expectations, the company must offer a convenient service [START_REF] Berry | Understanding Service Convenience[END_REF][START_REF] Liang | The behavioural sequence of the financial services industry in Taiwan : Service quality, relationship quality and behavioural loyalty[END_REF]. Convenience enables companies to strengthen their relationships with consumers and to amass competitive advantages (Seiders et al., 2007). When other companies also offer convenient services, the company must differentiate itself by offering better value if it is to maintain its market position [START_REF] Colwell | Toward a measure of service convenience : Multiple-item scale development and empirical test[END_REF]. Many scholars have found that consumer purchasing behavior is strongly influenced by service convenience [START_REF] Dai | Does service convenience matter? An empirical assessment of service quality, service convenience and exchange relationship in electronic mediated environment[END_REF].

Theoretical foundation: service convenience as a multidimensional construct

In this paper, we distinguish between two types of convenience in the marketing literature: service convenience [START_REF] Berry | Understanding Service Convenience[END_REF][START_REF] Brown | Convenience in Services Marketing[END_REF][START_REF] Gehrt | The dimensionality of the convenience phenomenon : A qualitative reexamination[END_REF] and retail convenience [START_REF] Beauchamp | Perceptions of Retail Convenience for in-store and Online Shoppers[END_REF][START_REF] Colwell | Toward a measure of service convenience : Multiple-item scale development and empirical test[END_REF][START_REF] Reimers | Retail centres : It's time to make them convenient[END_REF]Seiders et al., 2007). In the context of services, convenience is often associated with time or place, use, execution, and acquisition. Retail convenience is often described by reference to the dimensions of access, search, possession, transaction, utility, and post-use. In light of the absence of a scale that measures perceived convenience across the delivery process comprehensively, the purpose of the present study is to identify the dimensions of HD convenience. Accordingly, we develop the HDCCB scale as a parsimonious measurement tool. [START_REF] Gevaers | Characteristics of Innovations in Last-Mile Logistics-Using Best Practices, Case Studies and Making the Link with Green and Sustainable Logistics[END_REF] described several problems with HDs, one of which is the high proportion of failed deliveries. The quality of delivery is assessed primarily by reference to the availability of flexible delivery-time windows, reliability, ease of use, and trust in the delivery agent [START_REF] Ghajargar | Home delivery services : Innovations and emerging needs[END_REF]. Following [START_REF] Brown | Convenience in Services Marketing[END_REF] conceptualization of service convenience, we define HD convenience as the consumer's perceptions of product delivery in terms of time, use, execution, and acquisition.

Dimensions of home-delivery convenience

Time

We define the "time" dimension in relation to the expectation that products will be delivered within a reasonable timeframe, given a scheduled delivery window. Retailers have recognized that consumers see time as a valuable resource [START_REF] Beauchamp | Perceptions of Retail Convenience for in-store and Online Shoppers[END_REF]. A HD can be completed at a time that is convenient for the consumer. For example, courier shipments may be subject to specific constraints because companies allow customers to specify precise time windows [START_REF] Gruber | A new vehicle for urban freight? An ex-ante evaluation of electric cargo bikes in courier services[END_REF].

Use

When we write of "use," we mean to refer to the tracking of products during shipment and control over CO2 emissions as well as to the specificities of the delivery of sensitive products.

Planning and operations are critical to the quality of delivery services. E-retailers can offer additional services to facilitate the delivery of goods, such as more convenient delivery options.

Execution

In the context of the execution dimension, it is important to note that consumers expect to receive products in satisfactory condition and to be able to return them without undue difficulty [START_REF] Berry | Understanding Service Convenience[END_REF][START_REF] Brown | Convenience in Services Marketing[END_REF]. Consumers may be willing to outsource some (or all) of the work that is required to purchase certain products. The introduction of a service component (execution) into a product offering may make delivery more convenient. For example, some of the steps that must ordinarily be taken to select or return products may be eliminated.

Acquisition

In our framework, the term "acquisition" has to do with ensuring that a purchase can be completed without errors and through the use of a convenient payment method. Last-mile logistics service providers can make it easier, both financially and otherwise, for their customers to purchase their services. Accepting various payment methods, such as PayPal and credit cards, and the development of credit plans are salient examples. In addition, the use of GPS and tracking systems enables the dispatch of real-time notifications. This feature can make it easier to avoid failed deliveries and to manage accidents.

Methodology

As noted previously, the objective of our research is to identify dimensions that are related to the convenience of HD with CBs and to create a scale for measuring them. We developed the scale by following the procedures of Churchill (1979) and [START_REF] Devellis | Scale Development : Theory and Applications[END_REF], as well as other similar studies [START_REF] Asiah Omar | Measuring service quality in retail loyalty programmes (LPSQual) : Implications for retailers' retention strategies[END_REF][START_REF] Bagdare | Measuring retail customer experience[END_REF][START_REF] Bhatt | Attractive and facilitating store atmospheric stimuli : Validating the scales[END_REF][START_REF] Kumar | How to measure post-purchase customer experience in online retailing? A scale development study[END_REF]. We combined deductive and inductive methods. The deductive method entails the development of a theoretical basis for understanding and defining the construct under observation from the literature. The inductive method involves the generation of items from the responses of individuals, which we obtained from semi-structured interviews [START_REF] Hinkin | A Review of Scale Development Practices in the Study of Organizations[END_REF]. In more specific terms, we identified the most important studies on the convenience of services in the existing literature (Table I). Accordingly, we define HD convenience as a multidimensional construct that has four components: time, use, execution, and acquisition. Furthermore, we conducted semi-structured interviews in order to develop items that are related to the four dimensions. Finally, we identified the main dimensions through exploratory factor analysis and confirmed the scale through confirmatory factor analysis.

Creation of the items

We used semi-structured interviews [START_REF] Amasiatu | The management of first party fraud in e-tailing : A qualitative study[END_REF][START_REF] Foster | Using scenarios to explore employee attitudes in retailing[END_REF] to develop items that measure the convenience of HD by CBs. This approach allows for in-depth questioningconsumers are encouraged to speak freely about any areas they feel are relevant to the topic under examination [START_REF] Saunders | Research Methods for Business Students[END_REF]. Six men and four women between the ages of 21 and 50 participated (Table 2). They were asked about their experiences with and perceptions of the convenience of HD by CB. The respondents were familiar with e-commerce and various delivery methods, such as HD and pick-up stations. We reached a saturation point after 10 interviews and determined that collecting further data would yield no novel insights [START_REF] Charmaz | Constructing Grounded Theory : A Practical Guide through Qualitative Analysis[END_REF]. We then coded an initial list of 23 items, and two researchers conducted a content analysis [START_REF] Gavard-Perret | Lexical analysis in Marketing : Discovering the contents of the message or recognizing the models of enunciation[END_REF]) by using NVivo. Thereafter, three faculty members with experience in the marketing of retail and services assessed the content validity of the list. They were asked to match and refine the items by reference to the four dimensions of convenience: time, use, execution, and acquisition. After the revision, 19 items were retained for factor analysis.

Insert Table 2 about here

Sample and data collection

In order to collect the data, we used an online questionnaire with 19 items that we assigned to the four dimensions. The respondents rated the items on a five-point Likert scale (1 = "strongly disagree," 5 = "strongly agree"). We used two filter questions to ensure that the respondents were familiar with e-commerce and various delivery methods, such as HD, drop-offs, and pickup points. We received a total of 416 validly completed questionnaires (Table 3). We note that 53% of the respondents were female, 55% were between the ages of 35 and 44, and 58% held a master's degree. Following [START_REF] Hair | Multivariate Data Analysis[END_REF], we determined that our sample would need to be at least five times larger than the number of variables. Accordingly, we retained 116 responses for exploratory factor analysis. We examined the remainder (300 responses) as part of the confirmatory factor analysis [START_REF] Asiah Omar | Measuring service quality in retail loyalty programmes (LPSQual) : Implications for retailers' retention strategies[END_REF][START_REF] Bagdare | Measuring retail customer experience[END_REF]Childs et al., 2020).

Insert Table 3 about here

Data analysis and results

We used SPSS 23 and AMOS 18 for both the exploratory factor analysis and the confirmatory factor analysis.

Exploratory factor analysis

We conducted an exploratory factor analysis in order to assess the adequacy of the factor model and to test the multidimensionality of the convenience of HD with CBs. We used the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy and Bartlett's test of sphericity. The KMO value, 0.875, is larger than 0.5, and the result from Bartlett's test of sphericity (c 2 = 1,124.787, df = 190, p £ 0.0001) is significant, indicating that the items from the scale are correlated with each other-the null hypothesis of no correlation can be rejected. Accordingly, the data that we collected is suitable for factor analysis.

We then applied principal component analysis and the varimax rotation method with Kaiser normalization to two ends, namely 1) to consider factors with eigenvalues in excess of one and 2) to remove low commonalities, cross-loadings, and low factor loadings (< 0.5). After five iterations, we removed three items with low cross-loadings ("few out-of-stock situations" (0.34), "easy to move within and find what you are looking for" (0.27), "lower taxes included in the price" (0.39)). The results reveal that there are four factors with eigenvalues larger than one, which explain 65.204% of the total variance. The items load strongly on the factors that they represent (between 0.618 and 0.889). The first factor, time convenience, explains the largest proportion (42.039%) of the total variance. It comprises three items that are related to delivery within a reasonable time, respect for promised dates and times, and the flexibility of the time window. The second factor, which we labeled use convenience, explains 9.144% of the variance. It measures the use of tracing and tracking during shipment, reductions in CO2 emissions, respect for privacy, and the safety of the delivery of fragile products. The third factor, execution convenience, accounts for 7.487% of the variance and has to do with returns policies, the convenience of the conditions for returns, and direct contact with the delivery agent. The fourth factor is acquisition convenience (6.534% of the total variance), which is related to the accuracy of deliveries and to payment methods.

We also calculated Cronbach's alpha in order to assess the internal consistency of the factors. The Cronbach's alpha values are as follows: 0.703 for time convenience, 0.755 for use convenience, 0.796 for execution convenience, and 0.654 for acquisition convenience. These values are acceptable when compared to the recommended range (0.7-0.6) for scale reliability in exploratory research [START_REF] Hair | Multivariate Data Analysis[END_REF][START_REF] Nunnally | Psychometric Theory[END_REF]. The results from the exploratory factor analysis are presented in Table 4.

Insert Table 4 about here

Confirmatory factor analysis

We conducted the confirmatory factor analysis by using the maximum likelihood method in order to verify the factor structure and to examine the reliability as well as the convergent and discriminant validity of the convenience-of-delivery scale for CBs. Since the standardized factor loadings for two items from use convenience and acquisition convenience are below the acceptable threshold of 0.5 (Table 5), we removed them [START_REF] Bagozzi | On the evaluation of structural equation models[END_REF]. The loadings for all of the 14 items range between 0.654 and 0.868. The values of the fit indices are above the minimum thresholds [START_REF] Bagozzi | On the evaluation of structural equation models[END_REF][START_REF] Hair | Multivariate Data Analysis[END_REF]. These include the indicators of absolute fit (c 2 = 120.13, df = 64; GFI = 0.944 (> 0.9); RMSEA = 0.054 (< 0.08); SRMR = 0.029 (< 0.04)), of incremental fit (CFI = 0.979 (> 0.95); TLI = 0.970 (> 0.95); NFI = 0.956 (> 0.9)), and of parsimony fit (AGFI = 0.908 (> 0.9); c 2 /df = 1.877 (< 3), p £ 0.0001). The indices therefore confirm the adequate fit of the measurement model.

Insert Table 5 about here

Tests of reliability and validity

Cronbach's alpha and composite reliability (CR) are used to assess the reliability of the internal consistency of latent factors. The exploratory factor analysis indicates that the Cronbach's alpha values for the four factors range from 0.654 to 0.796 (Table 4). The confirmatory factor analysis indicates that CR takes the following values: 0.844 for time convenience, 0.808 for use convenience, 0.866 for execution convenience, and 0.710 for acquisition convenience (Table 5). All of these values are above the threshold of 0.7 [START_REF] Hair | Multivariate Data Analysis[END_REF], which demonstrates the adequate reliability of the measurement instrument. We then assessed convergent and discriminant validity. According to [START_REF] Fornell | Evaluating Structural Equation Models with Unobservable Variables and Measurement Error[END_REF], convergent validity indicates the extent to which the assessment measures are correlated with other measures to which they should be related. Convergent validity obtains when average extracted variance (AVE) is larger than or equal to 0.5 [START_REF] Fornell | Evaluating Structural Equation Models with Unobservable Variables and Measurement Error[END_REF]. As shown in Table 5, the AVE values for each construct in this study are in excess of 50%.

The regression coefficients of the parameter estimates are statistically significant (Table 6), indicating that all observable indicators load significantly on their respective latent factors.

Evidently, the convergent validity of the measurement instrument is adequate. We examined discriminant validity by comparing the correlations and the square roots of the AVE scores for each of the pairwise constructs [START_REF] Fornell | Evaluating Structural Equation Models with Unobservable Variables and Measurement Error[END_REF]. Since none of the correlations exceed the square root of the AVE scores for each pairwise construct (Table 7), the discriminant validity of the measurement instrument was confirmed-the constructs measure distinct concepts.

Insert Table 6 about here

Insert Table 7 about here

Concluding remarks

The concept of convenience has been studied in services marketing [START_REF] Berry | Understanding Service Convenience[END_REF][START_REF] Brown | Convenience in Services Marketing[END_REF][START_REF] Gehrt | The dimensionality of the convenience phenomenon : A qualitative reexamination[END_REF]Morganti et al., 2014;[START_REF] Punakivi | Identifying the success factors in e-grocery home delivery[END_REF] and in retail marketing [START_REF] Beauchamp | Perceptions of Retail Convenience for in-store and Online Shoppers[END_REF][START_REF] Colwell | Toward a measure of service convenience : Multiple-item scale development and empirical test[END_REF][START_REF] Reimers | Retail centres : It's time to make them convenient[END_REF]Seiders et al., 2007). However, research on HD remains scarce. In addition, few studies in this domain focus on the perspective of consumers. As [START_REF] Jiang | Measuring consumer perceptions of online shopping convenience[END_REF] pointed out, convenience is a contextual concept. This study adds to the literature and contributes to the understanding of consumer perceptions of the convenience of last-mile delivery. In developing a parsimonious measurement tool, the HDCCB scale, we identified the important factors that explain the manner in which customers evaluate the convenience of home delivery.

HD convenience, a multidimensional construct, measures each facet of the four dimensions (time, use, execution, and acquisition). These dimensions explain 65% of the variance. Among them, the time dimension accounts for 42%, the largest proportion. Evidently, consumers perceive convenience primarily in terms of the flexibility that the available time windows afford to them, which accords with Ghajargar et al.'s (2016) findings. Therefore, home delivery via CBs meets consumers' expectations about speed, flexibility, and product traceability. CBs are a promising solution for companies and e-retailers that are eager to improve the quality of their services in last-mile logistics.

We were also motivated by the environmental issues that HD raises for last-mile logistics providers. Many consumers are looking for a more environmentally friendly means of consuming that does not entail forgoing certain goods and services, such as HD [START_REF] Schliwa | Sustainable city logistics-Making cargo cycles viable for urban freight transport[END_REF]. Companies that use CBs to make HDs can benefit from the brand image of an environmentally friendly business that is committed to sustainability, which could generate a competitive advantage. E-retailers and last-mile logistics providers could use the measurement tool that we proposed to evaluate consumer perceptions of the convenience of HD.

Limitations

The study focused on consumer perceptions of the convenience of the last-mile delivery of goods on CBs by identifying, developing, and measuring the dimensions of convenience.

Psychological and contextual factors might also be important to the choice of delivery method, but we did not consider them. In addition, we only examined online experiences, and we did not discriminate between types of products. Nevertheless, the results are generalizable to all retailers with an online presence. Finally, the research was conducted in France and with French consumers. The replication of this study in contexts other than European city centers could yield interesting results, especially given that consumers' perceptions vary with their sociodemographic and cultural characteristics.

Directions for future research

Consumer perceptions of convenience are influenced by sociodemographic and cultural factors [START_REF] Bergadaa | The Role of Time in the Action of the Consumer[END_REF][START_REF] Durrande-Moreau | Time Styles and the Waiting Experience : An Exploratory Study[END_REF], which highlights the importance of testing our proposed scale in different countries and of considering the influence of variables such as age and gender. Another possibility would be to develop a convenience scale that refers to CB energy sources other than electricity. Batteries remain environmentally problematic due to the difficulty of destroying them once they are no longer usable (Mancini, 2017). Energy sources such as hydrogen can yield superior results for HD, leading to more favorable findings about the convenience variables. In addition, the use of renewable and thus more sustainable energy sources, such as solar and wind energy, could also produce superior results. Louis, D., Lombart, C., & Durif, F. (2019). Impact of a retailer's CSR activities on consumers' loyalty. International Journal of Retail & Distribution Management,47(8), https://doi.org/10.1108/ IJRDM-12-2018-0262 Ma, B., Wong, Y. D., & Teo, C.-C. (2022). Parcel self-collection for urban last-mile deliveries :

A review and research agenda with a dual operations-consumer perspective.

Table by authors

Table I . Dimensions of convenience in previous studies

 I

	Source		Dimensions of convenience	Context
	Brown (1990)	Time/place, acquisition, use, execution and convenience	Services in general
	Gehrt & Yale (1993)	Time, place, and effort		Services in general
	Transportation https://doi.org/10.1016/j.trip.2022.100719 Research Interdisciplinary Berry et al., (2002); Colwell et al., (2008); Seiders et al., (2007) Decision, access, transaction, benefits, and post-Perspectives, benefit convenience	16, Services in retail 100719.
	Mancini, S. (2017). The Hybrid Vehicle Routing Problem. Transportation Research Part C: Clulow & Reimers (2009); Reimers & Clulow (2009) Time, car, spatial, hedonic shopping, and effort convenience Retail centers
	Emerging Technologies, 78, 1-12. https://doi.org/10.1016/j.trc.2017.02.004 Manerba, D., Mansini, R., & Zanotti, R. (2018). Attended Home Delivery : Reducing last-mile Access, search, transaction, and possession Retail, in-store Beauchamp & Ponder (2010) convenience and online
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	deployment of pickup point networks in urban and suburban areas. Research in
	Transportation	Business	&	Management,	11,	23-31.

https://doi.org/10.1016/j.rtbm.2014.03.002

Table II . Sociodemographic description of the interviewees

 II

	N. Gender	Age (years)	Professional status	Experience (e-commerce platforms)
	1	Woman	50	Advertising director	Uber Eats, Delivroo
	2	Woman	34	Osteopath	Amazon, Veepee, Monopdelivery
	3	Woman	47	Nursery director	Amazon
	4	Woman	22	Student	Uber Eats
	5	Man	22	Student	Amazon, Asos
	6	Man	35	Wine merchant	Amazon
	7	Man	21	Student	Uber Eats, Amazon
	8	Man	23	Delicatessen seller	Amazon
	9	Man	50	Supply chain manager	Uber Eats, Deliveroo
	10	Man	31	Commercial agent	Uber Eats, Food Cheri

Table by authors Table III. Sample description

	Demographic factor		Frequency	Percentage
	Gender	Male	195	47
		Female	221	53
		Total	416	100
	Age	18 to 24	44	11
		25 to 34	69	17
		35 to 44	230	55
		45 to 54	56	13
		55 to 64	17	4
		Total	416	100
	Education	Bachelor	157	38
		Master	243	58
		Ph.D.	16	4
		Total	416	100
		Student	77	19
	Occupation	Employee	21	5
		Retailer	59	14
		Senior	259	62
		Executive
		Total	416	100
	Table by authors

Table IV . Results of the EFA

 IV

		Personnel contact with the
		delivery guy who is willing to	0.706
		help
		The delivery guy acts as an
		ambassador of the product or	0.807
		retail brand
	Acquisition	Accuracy, error-free sales
	convenienc	transactions, and home	0.889	1.045	6.534	0.654
	e	delivery records
		The payment terms seem correct	0.731
		Mobile payment (QR Code)	0.823
		Offer on the next delivery (E.g., voucher)	0.689
	Overall scale				65.204	0.886
	Table by authors
	Factors	Items	Factor loadings	Eigenv -alues	% of variance	Cronbach' s alpha
	Time
	convenienc	Delivery in a reasonable time 0.745	6.726	42.039	0.703
	e
		Respect for promised date and time	0.736
		Flexibility of delivery time window	0.618
	Use convenienc e	Tracing and tracking during shipment	0.700	1.463	9.144	0.755
		Fewer emissions of CO2	0.641
		Respect of privacy	0.625
		Safety of delivery of fragile products	0.657
	Execution
	convenienc	Products received undamaged 0.772	1.198	7.487	0.796
	e
		Ease of return products	0.741
		Return policy and conditions are convenient	0.684

Table V . Measurement model results Factors Items

 V Items with loadings below the acceptable threshold of 0.5 were removed

			Factor
			loading	CR	AVE
			s
	Time convenience	(1) Delivery in a reasonable time	0.826	0.844 0.643
		(2) Respect of (promised date and time	0.815
		(3) Flexibility of delivery time window	0.754
	Use convenience	(1) Tracing and tracking during	0.731	0.808 0.513
		shipment
		(2) Respect of privacy*	0.44
		(3) Fewer emissions of CO2	0.693
		(4) Safety of delivery of fragile	0.743
		products
	Execution	(1) Products received undamaged	0.868	0.866 0.567
	convenience
		(2) Ease of returning products	0.740
		(3) Return policy and conditions are	0.743
		convenient
		(4) Personnel contact with the delivery	0.654
		guy who is willing to help
		(5) The delivery guy acts as an	0.807
		ambassador of the product or retail
		brand
	Acquisition	(1) Accuracy, error-free sales	0.790	0.710 0.534
	convenience	transactions, and home delivery records
		(2) The payment terms seem correct	0.708
		(3) Mobile payment (QR Code)	0.692
		(4) Offer on the next delivery (e.g.,	0.38
		voucher)*

*

Table by authors Table VI. Parameter estimates -regression weights

		Estimate SE	CR	P
	Time3 <---Time	0.944 0.065 14.483	***
	Time2 <---Time	0.986 0.059 16.664	***
	Time1 <---Time	1.000
	Use3 <---Use	1.000
	Use2 <---Use	0.929 0.075 12.407	***
	Use1 <---Use	0.962 0.073 13.168	***
	Exec5 <---Execution	1.000
	Exec4 <---Execution	0.929 0.077 12.116	***
	Exec3 <---Execution	0.980 0.063 15.570	***
	Exec2 <---Execution	0.948 0.067 14.219	***
	Exec1 <---Execution	0.986 0.063 15.745	***
	Acq3 <---Acquisition	0.359 0.109 3.280	.001
	Acq2 <---Acquisition	0.995 0.074 13.379	***
	Acq1 <---Acquisition	1.000
	Note: ***p value significant at the level 0.0001

Table by authors Table VII. Convergent and discriminant validity

  Notes: The square root of AVE scores are represented on the diagonal, the correlations coefficients between constructs are represented below the diagonal.

		Mean SD	1	2	3	4
	(1) Time (3 items)	4.07 0.80	0.80
	(2) Use (3 items)	4.06 0.89	0.55	0.71
	(3) Execution (5 items)	4.00 0.87	0.58	0.60 0.75
	(4) Acquisition (3 items)	3.48 0.77	0.39	0.51 0.49	0.73

in the retail apocalypse : Investigating consumers' buying behaviours. International Journal of Retail & Distribution Management, 48(3), 262-286. https://doi.org/10.1108/IJRDM-03-2019-0079 Churchill, G. A. (1979). A Paradigm for Developing Better Measures of Marketing Constructs. Journal of Marketing Research, 16(1), 64-73. https://doi.org/10.1177/002224377901600110 Clulow, V., & Reimers, V. (2009). How do Consumers Define Retail Centre Convenience? Australasian Marketing Journal,

Seiders, K., Voss, G. B., Godfrey, A. L., & Grewal, D. (2007). SERVCON : Development and validation of a multidimensional service convenience scale. Journal of the Academy of Marketing Science,35(1), 144-156. https://doi.org/10.1007/s11747-006-0001-5 Sheth, M., Butrina, P., Goodchild, A., & McCormack, E. (2019)


P Comon

J M F Ten Berge

L De Lathauwer

J Castaing

Generic and Typical Ranks of Multi-Way Arrays

Keywords: Tensor, Multiway Arrays, Generic Rank, Canonical Decomposition, Parafac, Factor Analysis AMS Classification: 15A69 15A03 15A72 15A57 46M05

The concept of tensor rank was introduced in the twenties. In the seventies, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini's lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries.

Introduction

Generic ranks, defined in the complex field, have been studied for several decades [START_REF] Howell | Global properties of tensor rank[END_REF] [START_REF] Strassen | Rank and optimal computation of generic tensors[END_REF]. However, the value of the generic rank for arbitrary dimensions is not yet known in the unsymmetric case, and has been known in the symmetric case only recently [START_REF] Alexander | Polynomial interpolation in several variables[END_REF] [4] [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF]. The existence itself of the generic rank is not ensured in the real case, and there exist in general several typical ranks (see section 2.1 for definitions). The typical tensor rank of three-way arrays over algebraically closed fields has been much studied in the context of computational complexity theory. Bürgisser, Clausen and Shokrollahi [START_REF] Ürgisser | Algebraic complexity theory[END_REF]Ch.20] give an overview of general results for various classes of arrays; these results have been extended in [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF] in the symmetric case. The study of tensor rank over the real field has lagged behind. In this paper, generic and typical ranks are discussed for various tensor structures.

The typical rank of three-way arrays over the real field has been relevant for psychological data analysis since Carroll and Chang [START_REF] Carroll | Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition[END_REF] and Harshman [START_REF] Harshman | Foundations of the Parafac procedure: Models and conditions for an explanatory multimodal factor analysis[END_REF] independently proposed a method which they christened Candecomp and Parafac, respectively. Therefore, we shall subsequently refer to this decomposition with the acronym CP, as several other authors did before, even if the CP decomposition had been introduced much earlier [START_REF] Hitchcock | Multiple invariants and generalized rank of a p-way matrix or t ensor[END_REF]. This CP decomposition generalizes Principal Component Analysis to three-way data, by seeking the best least squares approximation of a data array by the sum of a limited number of rank-one arrays. In 2-way analysis, the rank of the data matrix is the maximum number of components that Principal Component Analysis can extract, up to scale and rotation ambiguities. This property generalizes smoothly to three-way data. That is, the rank of a three-way array is the maximum number of components that CP can extract, up to scale and permutation ambiguities. Thus, the study of typical rank of three-way arrays is of great theoretical importance for CP.

Although CP was developed in a psychometric environment, its main area of applications has been Chemometrics, e.g. [START_REF] Smilde | Multi-Way Analysis[END_REF]. In addition to straightforward Email addresses: pcomon@unice.fr (P. Comon), J.M.F.ten.Berge@rug.nl (J.M.F. ten Berge), Lieven.DeLathauwer@kuleuven-kortrijk.be (L. De Lathauwer). application of CP, chemometricians also use Tucker3 component analysis [START_REF] De Lathauwer | A multilinear singular value decomposition[END_REF] [12] [START_REF] Tucker | Some mathematical notes for three-mode factor analysis[END_REF] quite often. This is another model aiming at decomposing a data array as weighted sum of rank-one arrays, the weights being collected in a socalled core array. Typically, the underlying chemometric model dictates that a vast majority of specified core elements vanish. Because there exist admissible transformations which generate a vast majority of zero elements in arbitrary arrays, we need tools to tell models from tautologies. This is where the concept of typical rank has found another realm of application. For instance, Ten Berge and Smilde [START_REF] Smilde | Non-triviality and identification of a constrained tucker 3 analysis[END_REF] have argued that a sparse core hypothesized by Gurden et al. [START_REF] Gurden | Modeling of spectroscopic batch process data using grey models to incorporate external information[END_REF] is indeed a model and not a tautology. Their hypothetical core was a 5×5× 3 array with only 5 nonzero entries, hence of rank 5 at most. Because 5 × 5 × 3 arrays have a typical rank of at least 7, it is clear at once that transformations which yield as few as five non-zero elements, starting from any randomly generated 5×5×3 array, do not exist. In this way, the typical rank of three-way arrays finds applications in distinguishing constrained Tucker3 models from tautologies. Besides Chemometrics, CP has found important applications in signal processing, especially in Independent Component Analysis [16] [17] and in multi-user access in wireless communications [START_REF] Sidiropoulos | Blind parafac receivers for ds-cdma systems[END_REF] [19] [START_REF] Sidiropoulos | Khatri-rao space-time codes[END_REF]. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality [21, p. 125] [22] [23].

The paper is organized as follows. In section 2, definitions and historical remarks are provided. Next, a numerical algorithm is described in section 3, which is able to compute the generic rank of any tensor, symmetric or not. This approach is based on the so-called Terracini's lemma. Numerical values are reported in section 4, and compared to the already known rank values previously obtained by means of algebraic calculations. The consistency of the results confirm the validity of the approach, which can yield generic ranks for more complicated structures, such as tensors with symmetric matrix slices, which occur in the context of the Indscal model [START_REF] Carroll | Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition[END_REF], among others.

Generic and Typical Ranks

Definitions

Let T be a L-way array of dimensions N ℓ , 1 ≤ ℓ ≤ L, with values in a ring R. This array always admits a decomposition into a sum of outer products as:

T = R r=1 u (1) r • u (2) r • . . . • u (L) r (1)
where u (ℓ) r is a vector of dimension N ℓ , ∀r, and • denotes the tensor product2 .

Now consider an array T with values in a field K. Arrays u (ℓ) r may be considered as vectors of the linear space K N ℓ . Thus, as a combination of tensor products of vectors, T may be considered as a tensor. Under a linear change of coordinate system in each space K N ℓ , defined by a matrix A (ℓ) , the tensor is represented by another array, obtained by the multi-linear transform {A (1) , A (2) , . . . , A (L) }. Since it is legitimate once a basis has been defined in the space, no distinction will be made in the remainder between the tensor and its array representation.

The rank of a given tensor T (and by extension, of the array defining its coordinates in a given basis) is the minimal integer R such that the decomposition (1) is exactly satisfied. This decomposition is referred to as the tensor Canonical or Parallel factor Decomposition (CP).

A property is called typical if it holds true on a set of nonzero volume [START_REF] Strassen | Rank and optimal computation of generic tensors[END_REF] [24] [25] [5]. This supposes that some topology has been defined on K N 1 ×N 2 ×...N L ; this can be the Zariski topology for instance, or an Euclidean topology. A property is said to be generic if it is true almost everywhere. In other words, a generic property is typical, but the converse is not true.

Let N 1 , . . . , N L be given positive integers. Then the rank of tensors of size

N 1 × N 2 × • • • × N L is
bounded, and one can make a partition of the tensor space, according to the rank values. One can define typical ranks as the ranks that are associated with subsets of nonzero volume in the latter partition. If there is a single typical rank, then it may be called the generic rank.

For instance, there is a single generic rank if the underlying field K is algebraically closed (as the field of complex numbers, C) [2] [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF]. But there may be several typical ranks if K is the real field, R. In the complex field, the calculation of the generic rank of symmetric tensors is completely described by the Alexander-Hirschowitz (AH) theorem [START_REF] Alexander | Polynomial interpolation in several variables[END_REF]. Recently, Abro and Ottaviani attempted to generalize the AH theorem to non symmetric complex tensors [START_REF] Abo | Induction for secant varieties of Segre varieties[END_REF], and provided an almost exhaustive list of exceptions. This recent contribution is the most significant step towards the complete characterization of the generic rank of unsymmetric tensors with free entries.

Historical remarks

Bounds on the typical rank over the complex field were given in [START_REF] Strassen | Rank and optimal computation of generic tensors[END_REF]. The study of typical rank over the real field was initiated by Kruskal [START_REF] Kruskal | Statement of some current results about three-way arrays[END_REF] [24], who noted that 2 × 2 × 2 arrays had both rank 2 and rank 3 with positive probability. Kruskal also added a few typical ranks for small arrays. Franc [START_REF] Franc | Etudes algébriques des multitableaux: Apports de l'algèbre tensiorielle[END_REF] discussed some more results, including bounds on typical rank. Ten Berge and Kiers [START_REF] Bolasco | Simplicity of core arrays in three-way principal component analysis and the typical rank of p × q × 2 arrays[END_REF] gave a first result of some generality, in solving the typical rank issue for all two-slice arrays (that is, arrays of format 2 × N 2 × N 3 ). These results were further generalized in [START_REF] Berge | The typical rank of tall three-way arrays[END_REF], to include all cases where, for

N 1 ≥ N 2 ≥ N 3 , N 1 > N 2 N 3 -N 2 .
Additional miscellaneous results can be found in [START_REF] Atkinson | Bounds on the ranks of some 3-tensors[END_REF] [START_REF] Berge | The typical rank of tall three-way arrays[END_REF] [31] [START_REF] Berge | Symmetry transformations for square sliced three way arrays, with applications to their typical rank[END_REF].

When Carroll and Chang developed Candecomp, the main applications they had in mind (a scalar product fitting problem related to Indscal) involved three-way arrays with slices that are symmetric in two of the three modes. Ten Berge, Sidiropoulos and Rocci [START_REF] Berge | Typical rank and INDSCAL dimensionality for symmetric three-way arrays of order i × 2 × 2 or i × 3 × 3[END_REF] noted that this form of symmetry would affect typical ranks, and examined a number of cases; also see [START_REF] Berge | The Carroll & Chang conjecture of equal INDSCAL components when Candecomp/Parafac gives perfect fit[END_REF]. Quite often, indeed, symmetry of slices appears to entail lower typical rank values. On the other hand, there are also cases where symmetry of the slices does not affect the typical rank. A partial explanation for this can be found in [START_REF] Berge | Symmetry transformations for square sliced three way arrays, with applications to their typical rank[END_REF]. Ten Berge et al. also noted that symmetric slices are often double centered [7, p. 286], which will further reduce the typical rank. That is, when an array has N 1 double centered slices of order N 2 × N 2 , it can be reduced to a N 1 × (N 2 -1) × (N 2 -1) array, and its typical rank will therefore be the same as that of noncentered symmetric-slice N 1 × (N 2 -1) × (N 2 -1) arrays. A parallel reasoning can be carried out for any nonsymmetrical double centered slices. A rationale for double centering slices in the Parafac context can be found in [35, p. 239]. It is easy to show that the N 1 × N 2 × N 3 array with N 1 double centered slices has the same typical rank as the uncentered N 1 × (N 2 -1) × (N 3 -1) array.

Computation of Generic Ranks

The algorithm proposed is directly inspired by [START_REF] Comon | Decomposition of quantics in sums of powers of linear forms[END_REF], which is in turn based on the so-called Terracini's lemma [START_REF] Terracini | Sulla rappresentazione delle forme quaternarie mediante somme di poten ze di forme lineari[END_REF] [37] [START_REF] Wakeford | Apolarity and canonical forms[END_REF]; note that the latter is often attributed to Lasker, and is hence almost one hundred years old. In a few works, the principle is based on the fact that the dimension of an irreducible variety is equal almost everywhere to the dimension of its tangent space (and we know that the set of tensors of rank at most R is irreducible [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF], for any R). Thus the dimension of a variety can be computed by measuring the dimension of the tangent space at a generic3 point. Let's see now how to compute the tangent space of interest. Equation ( 1) can be seen as a parameterization of tensor T . In fact, given a set of vectors {u (ℓ) r ∈ K N ℓ , 1 ≤ ℓ ≤ L, 1 ≤ r}, consider the mapping ϕ defined from a known subspace T R of (K

N 1 × K N 2 × • • • × K N L ) R onto K N 1 N 2 ... N L as: {u (ℓ) r ∈ T R , 1 ≤ ℓ ≤ L, 1 ≤ r ≤ R} → R r=1 u (1) r • u (2) r • . . . • u (L) r Denote Z R = ϕ(T R
) the image of this mapping. Then the dimension D of its closure ZR is given by the rank of the Jacobian of ϕ taken at a generic point, expressed in any fixed basis of K N 1 N 2 ... N L . If the Jacobian is of maximal rank at a generic point, that is, if its rank equals almost everywhere the dimension of the image space (e.g. N 1 N 2 . . . N L for unconstrained arrays), then it means that R is a typical rank. Actually, R will be either the smallest typical rank, or the generic rank. Note that it is always possible to reach the maximal Jacobian rank by increasing the number of terms R, so that the smallest4 typical rank is always found.

This result yields the following numerical algorithm:

• Express formally the parameterized rank-one tensor term in a canonical basis • Express formally the gradient of the latter in this basis • Draw randomly the parameters according to an absolutely continuous distribution, and initialize matrix J with the numerical value of the gradient • R = 1 • While rank(J ) strictly increases, do:

• Draw randomly the parameters according to an absolutely continuous distribution, and append this new numerical value of the gradient as a new row block in J • Compute the new value of D = rank(J )

• R ← R + 1
• Compute the dimension of the fiber of solutions as F = M -D, the difference between the number of parameters and the dimension of the image ZR .

In order to clarify the description of this algorithm, we give now the exact expressions of the Jacobian in various cases.

Jacobian for 3rd order asymmetric tensors with free entries

The mapping takes the form below

{a(r), b(r), c(r)} ϕ -→ T = R r=1 a(r) • b(r) • c(r)
taking into account the presence of redundancies, the number of parameters in this parameterization is

M = R(N 1 + N 2 + N 3 -2)
. In a canonical basis, T has the coordinate vector:

R r=1 a(r) ⊗ b(r) ⊗ c(r)
where we may decide that a(r), b(r), and c(r) are row arrays of dimension N 1 , N 2 , and N 3 , respectively, and ⊗ denotes the Kronecker product. Hence, after R iterations, the Jacobian of ϕ is the R(

N 1 + N 2 + N 3 ) × N 1 N 2 N 3 matrix: J =                                     I N 1 ⊗ b(1) ⊗ c(1) a(1) ⊗ I N 2 ⊗ c (1) a(1
I N 1 ⊗ b(R) ⊗ c(R) a(R) ⊗ I N 2 ⊗ c(R) a(R) ⊗ b(R) ⊗ I N 3                                     (2)
The values of the generic rank obtained with this algorithm, called rangj3(N1,N2,N3), or rangj(N,L) 5 for tensors of arbitrary order L and equal dimensions, are reported in tables 1, 2, and 3.

Jacobian for 3rd order asymmetric tensors with symmetric matrix slices

In this section, we consider tensors of size N 2 × N 2 × N 3 , having symmetric N 2 × N 2 matrix slices. We consider the mapping:

{b(r), c(r)} -→ T = R r=1 b(r) • b(r) • c(r).
Our code rgindscal3(N2,N3) implements the computation of the rank of the Jacobian below, when its size increases according to the algorithm described in section 3:

J =               I N 2 ⊗ b(1) ⊗ c(1) + b(1) ⊗ I N 2 ⊗ c(1) b(1) ⊗ b(1) ⊗ I N 3 . . . I N 2 ⊗ b(R) ⊗ c(R) + b(R) ⊗ I N 2 ⊗ c(R) b(R) ⊗ b(R) ⊗ I N 3               (3)
After R iterations, this matrix is of size

R(N 2 + N 3 ) × N 2 2 N 3 .
The number of parameters in this parameterization is M = R(N 2 + N 3 -1). Values of the generic rank are reported in table 4.

Jacobian for 3rd order double centered tensors with symmetric matrix slices

Now, take again N 2 × N 2 × N 3 tensors with symmetric N 2 × N 2 matrix slices, but assume in addition that every row and column in the latter matrix slices are zero-mean. In order to achieve this, it is sufficient to generate vectors b(r) with zero mean [START_REF] Comon | Generic and typical ranks of threeway arrays[END_REF]; in other words, only N 2 -1 random numbers need to be drawn, the last entry of each vector b(r) being obtained via b

N 2 = -N 2 -1 n 2 =1 b n 2 .
The Jacobian takes then the expression below:

J =               [I N 2 -1 , -1] ⊗ b(1) ⊗ c(1) + b(1) ⊗ [I N 2 -1 , -1] ⊗ c(1) b(1) ⊗ b(1) ⊗ I N 3 . . . [I N 2 -1 , -1] ⊗ b(R) ⊗ c(R) + b(R) ⊗ [I N 2 -1 , -1] ⊗ c(R) b(R) ⊗ b(R) ⊗ I N 3               (4)
where 1 denotes a column of ones of size N 2 -1. At the Rth iteration, this matrix is of size R(N 2 + N 3 -1) × N 2 2 N 3 . The number of parameters in this parameterization is M = R(N 2 + N 3 -2). Table 5 reports some numerical values obtained with the code rgindscal2z.

Jacobian for 3rd order tensors with double centered matrix slices

The previous reasoning can be applied to N 1 × N 2 × N 3 tensors with no symmetry constraint and whose N 1 × N 2 matrix slices have zero-mean rows and columns. As before, it is sufficient to generate vectors a(r) and b(r) with zero mean. The Jacobian is then equal to:

J =                      [I N 1 -1 , -1] ⊗ b(1) ⊗ c(1) a(1) ⊗ [I N 2 -1 , -1] ⊗ c(1) a(1) ⊗ b(1) ⊗ I N 3 . . . [I N 1 -1 , -1] ⊗ b(R) ⊗ c(R) a(R) ⊗ [I N 2 -1 , -1] ⊗ c(R) a(R) ⊗ b(R) ⊗ I N 3                      (5)
At the Rth iteration, this matrix is of size

R(N 1 + N 2 + N 3 -2) × N 1 N 2 N 3 .
The number of parameters in this parameterization is M = R(N 1 + N 2 + N 3 -3). The numerical values obtained with the code rangj3z are not reported, since we always have, for any triplet (N 1 , N 2 , N 3 ): rangj3z(N1,N2,N3)=rangj3(N1-1,N2-1,N3). In other words, as far as the generic rank is concerned, centering in a given mode of dimension N i yields the same effect as reducing the dimension to N i -1, which makes sense.

Jacobian for symmetric tensors

In the case of symmetric tensors of dimension N and order L, the mapping ϕ is defined from K N R to the space of symmetric tensors [START_REF] Comon | Decomposition of quantics in sums of powers of linear forms[END_REF], or equivalently to K p with p = N +L-1 L , as:

{a(r) ∈ K N , 1 ≤ r ≤ R} ϕ -→ R r=1 a(r) • L
where • stands for the tensor (outer) product; once a basis is chosen, the tensor product may be expressed by a Kronecker product, yielding a similar expression. In the case of order-3 tensors (L = 3) and after R iterations, the Jacobian of ϕ takes the following form, somewhat simpler than the previous cases:

J =        I N ⊗ a(1) ⊗ a(1) + a(1) ⊗ I N ⊗ a(1) + a(1) ⊗ a(1) ⊗ I N . . . I N ⊗ a(r) ⊗ a(r) + a(r) ⊗ I N ⊗ a(r) + a(r) ⊗ a(r) ⊗ I N        (6)
This matrix is of size RN × N 3 , but we know that its rank cannot exceed

N +2 3
= N(N + 1)(N + 2)/6. The number of parameters in this parameterization is M = RN. Numerical values of the generic rank obtained with rangjs(N,L) are reported in table 6.

Jacobian for 4th order tensors with Hermitian symmetry

In this section, we consider fourth-order tensors of size N × N × N × N with Hermitian symmetry. We say that a fourth-order tensor T has Hermitian symmetry if it satisfies:

T (i, j, k, l) = T (k, j, i, l) = T (i, l, k, j) = T (j, i, l, k) * (7)
for all values of indices i, j, k and l. In particular, fourth-order complex circular cumulants have this Hermitian symmetry. The space of fourth-order tensors with Hermitian symmetry is denoted by S(N). We consider the following mapping from C N R to S(N):

{v(r) ∈ C N , 1 ≤ r ≤ R} ϕǫ -→ R r=1 ǫ r v(r) ⊗ v(r) * ⊗ v(r) ⊗ v(r) *
in which ǫ r = ±1.

A symmetric real tensor has N(N + 1)(N + 2)(N + 3)/24 degrees of freedom. However, in the complex case, a tensor enjoying Hermitian symmetries (7) has N 2 (N + 1) 2 /4 distinct entries, of which some are strictly real. By counting the number of different real and imaginary parts of the tensor entries, we obtain that S(N) is a real vector space of dimension K(N) [40, Annexe B]:

K(N) = 6 N 4 + 4 N 1 N -1 2 + 3 N 2 + 2 N 1 N -1 1 + N 1 . (8)
It is formed by the union of the images of ϕ ǫ , for all ǫ = (ǫ 1 , . . . , ǫ r ). Note that ǫ l does not affect the rank of the Jacobian. For clarity it will be omitted in the expressions below.

Function ϕ ǫ is not analytic in C. However, it is differentiable in R. The Jacobian of ϕ ǫ , interpreted as a mapping from R 2N R to S(N), now takes the following form:

J =               Re(γ re (v(1))) Im(γ re (v(1))) Re(γ im (v(1))) Im(γ im (v(1))) . . . . . . Re(γ re (v(R))) Im(γ re (v(R))) Re(γ im (v(R))) Im(γ im (v(R)))               (9)
with

γ re (v(r)) = I N ⊗ v(r) * ⊗ v(r) ⊗ v(r) * + v(r) ⊗ I N ⊗ v(r) ⊗ v(r) * + v(r) ⊗ v(r) * ⊗ I N ⊗ v(r) * + v(r) ⊗ v(r) * ⊗ v(r) ⊗ I N and γ im (v(r)) = iI N ⊗ v(r) * ⊗ v(r) ⊗ v(r) * + v(r) ⊗ -iI N ⊗ v(r) ⊗ v(r) * + v(r) ⊗ v(r) * ⊗ iI N ⊗ v(r) * + v(r) ⊗ v(r) * ⊗ v(r) ⊗ -iI N .
The matrix J is of size 2RN × 2N 4 , but we know that its rank cannot exceed K(N). The number of parameters in this parameterization is M = 2RN. Numerical values of the smallest typical rank obtained with rangjh4(N) are reported in table 7.

Numerical results

The available results on unconstrained, slicewise symmetric, and double centered arrays can be compared with the numerical values delivered by the computer codes.

Tensors with free entries. Table 1 reports typical ranks for 2-slice, 3-slice, and 4-slice arrays. All known typical rank values [START_REF] Berge | The typical rank of tall three-way arrays[END_REF][START_REF] Berge | Partial uniqueness in CANDECOMP/PARAFAC[END_REF][START_REF] Berge | Symmetry transformations for square sliced three way arrays, with applications to their typical rank[END_REF] are reported in plain face, and are compatible with the results from rangj3. Specifically, the smallest of the known typical rank values within a cell coincides throughout with the results from rangj3. For the yet unknown entries, we insert the results from rangj3 in bold face. These bold face values represent the smallest typical rank in the real field, and the generic rank in the complex field. 1 Typical ranks for 2-slice, 3-slice, and 4-slice unconstrained arrays. Values reported in bold correspond to smallest typical ranks computed numerically; values in plain font were known before. Values separated by commas are known typical ranks. In the complex field, the smallest value in a cell is generic.

The values reported in table 1 correspond to necessary conditions that ensure uniqueness of the CP, in the sense that a tensor having a rank larger than those generic values has infinitely many CPs in C. Conversely, if the rank of a tensor is smaller than these generic values, then there is almost surely a finite number6 of possible CPs up to a scale factor, in either R or C. On the other hand, the bound given by Kruskal [START_REF] Kruskal | Rank, decomposition, and uniqueness for 3-way and n-way arrays[END_REF] corresponds to a sufficient condition ensuring uniqueness of the CP up to a permutation, and is always smaller than or equal to (but generally much smaller than) the generic rank. Other sufficient conditions have been derived in the literature under somewhat different assumptions, see e.g. [41] [42] for tensors enjoying Hermitean symmetries.

We report values of the smallest typical/generic rank of 3-way arrays with equal dimensions in table 2. The values shown in table 2 can also be compared to those obtained in the symmetric case (see table 6). Now the algorithm can be run on tensors of order higher than 3. To make the presentation of the results readable, table 3 rank obtained for asymmetric tensors with equal dimensions, N, and order L, with an algorithm referred to as rangj(N,L). These results are consistent with those of [START_REF] Lickteig | Typical tensorial rank[END_REF]. We also indicate the dimensionality of the fiber of solutions. This number is simply defined as the difference:

F (N, L) = R(N, L) (LN -L + 1) -N L .
For those values of dimension and order for which F = 0, only a finite number of different CPs are possible.

Tensors with symmetric matrix slices. Having verified that rangj3 and rangj work correctly throughout the cases where the generic/typical ranks are known, we next turn to the N 1 × N 2 × N 2 arrays with N 1 symmetric slices (Table 4). Again, known values coincide with numerical ones delivered by the code rgindscal3. We inserted results obtained from rgindscal3 alone in bold face. As far as can be determined, all results are again in agreement with previously known values [33] [34].

Tensors with double centered symmetric matrix slices. When the matrix slices are symmetric and also row-wise (or column-wise, which is the same thing) zero-mean, the code rgindscal2z yielded the values reported in table 5. Note that the generic rank computed by rgindscal2z(N2,N1) is the same as that computed by rgindscal3(N2-1,N1), at least according to the values explored in table 4. This shows that our method is easily adapted to handle the special case of double centered matrices.

Tensors with double centered matrix slices without symmetry constraint. A similar observation holds also true when the centered matrix slices are not symmetric. We do not separately report typical rank values for the case of double-centered (non symmetric) slices. Instead, we verified that the values obtained numerically with centering coincided with the values obtained numerically for uncentered arrays: rangj3z(N1,N2,N3)=rangj3(N1-1,N2-1,N3). Symmetric tensors. In table 6 we report values obtained with 3-way or 4way symmetric arrays, obtained with the code rangjs. Note that these results have been already reported in [START_REF] Comon | Decomposition of quantics in sums of powers of linear forms[END_REF]. The dimensionality of the fiber of solutions is given by:

F (N, L) = R N - N + L -1 L .
It is interesting to compare the ranks with those of the unsymmetric case, obviously larger, reported in table 3. In particular, one can observe that that the case F = 0 is again rarely met with generic arrays, but less rarely than in the non-symmetric case.

Fourth-order tensors with Hermitian symmetry. Finally, we report in table 7 values obtained for fourth-order tensors having Hermitian symmetry. These values were obtained with the code rangj4h.

1

 1 Part of this work has been presented at the TRICAP Workshop, Chania, Crete, Greece, June 4-9, 2006. It has been supported in part by the contract ANR-06-BLAN-0074 "Decotes", in part by (1) the Research Council K.U.Leuven, Belgium, under Grant GOA-Ambiorics, CoE EF/05/006 Optimization in Engineering (OPTEC), CIF1, STRT1/08/023, (2) F.W.O.: (a) project G.0321.06, (b) Research Communities ICCoS, ANMMM and MLDM, (3) the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, "Dynamical systems, control and optimization", 2007-2011), (4) EU: ERNSI. This publication only reflects the authors' views.

  ) ⊗ b(1) ⊗ I N 3 . . . I N 1 ⊗ . . . ⊗ . . . . . . ⊗ I N 2 ⊗ . . . . . . ⊗ . . . ⊗ I N 3 . . .

  typical rank R for N 1 × N 2 × N 2 arrays, with N 2 × N 2 symmetric slices having zero-mean columns. In the complex field, these values are generic. Smallest typical ranks R of symmetric arrays of dimension N and order L. In the complex field, these values are generic. (bottom) Number F of remaining degrees of freedom; when F = 0, there are only a finite number of CP solutions.

Table 2

 2 reports values of the generic Smallest typical rank R of unconstrained arrays of dimension N × N × N . Values reported in bold correspond to ranks computed numerically; values in plain font were known before. In the complex field, these values are generic.

	N 2 3 4 5	6	7	8	9
		R 2 5 7 10 14 19 24 30
	L N 2 3	4	5	6	7	8
	3	2 5	7 10 14 19 24
	4	4 9 20 37 62 97
	L N 2 3	4	5	6	7	8
	3	0 8	6	5	8	18 16
	4	4 0	4	4	6	24

Table 3 (

 3 top) Smallest typical rank R of unconstrained arrays of equal dimensions, N , and order L. In the complex field these values are generic. (bottom) Number F of remaining degrees of freedom; when F = 0, there are only a finite number of CP solutions.

Table 4

 4 Typical ranks for N 1 × N 2 × N 2 arrays, with N 2 × N 2 symmetric slices. Values reported in bold correspond to smallest typical ranks computed numerically. Values separated by commas are known typical ranks. In the complex field, the smallest value in a cell is generic.

	N 1 N 2	2	3	4	5
	2	2,3 3,4 4,5 5,6
	3	3	4	6	7
	4	3 4,5	6	8
	5	3 5,6	7	9
	6	3	6	7	9
	7	3	6	7	10
	8	3	6	8	10
	9	3	6 9,10 11
	10	3	6	10	11

Table 7

 7 Smallest typical rank R of fourth-order tensors with Hermitian symmetry of dimension N × N × N × N . Values reported in bold correspond to ranks computed numerically.

This notation is used in order to make the distinction with the matrix Kronecker product used in the next section.

In practice, a generic point is drawn randomly. If the result is questionable, the point can be drawn another time. Any random drawing should yield the same dimension, if the pdf is absolutely continuous.

Finding all typical ranks is still an open problem, and is not addressed in this paper.

The codes can be downloaded from www.i3s.unice.fr/∼pcomon.

This finite number of solutions may possibly be larger than the number of permutations. In such a case, the solution is not essentially unique.